lecture notes - simon fraser universityvjungic/calculus 1/calculus1.pdf · 2020. 7. 25. · 6.a...

Calculus I with Review

Differential Calculus

Lecture Notes

Veselin Jungic & Jamie MulhollandDepartment of Mathematics

Simon Fraser University

c©Jungic/Mulholland, August 8, 2018License is granted to print this

document for personal/educational use.

Contents

Contents i

Preface iii

Greek Alphabet v

1 Review: Functions and Models 1

1.1 Four Ways to Define a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Mathematical Models: A Catalog of Essential Functions . . . . . . . . . . . . . . . . . . . . . 9

1.3 New Functions From Old Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Exponential Functions & Inverse Functions and Logarithms . . . . . . . . . . . . . . . . . . . 23

Review: Preparation for Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Limits and Derivatives 37

2.1 The Tangent and Velocity Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2 The Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Calculating Limits Using the Limit Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4 The Precise Definition of Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6 Limits at Infinity: Horizontal Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Review: Problem Solving and Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.7 Derivatives and Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.8 The Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Differentiation Rules 87

3.1 Derivatives of Polynomials and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 88

3.2 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.4 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.5 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

i

ii CONTENTS

3.6 Derivatives of Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.7 Rates of Change in the Natural and Social Sciences . . . . . . . . . . . . . . . . . . . . . . . . 118

3.8 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Review: Preparation for Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.9 Related rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.10 Linear Approximation and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.11 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4 Applications of the Derivative 149

4.1 Maximum and Minimum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.2 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.3 How Derivatives Affect the Shape of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.4 Indeterminate Forms and L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.5 Summary of Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

4.6 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.7 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.8 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5 Parametric Equations and Polar Coordinates 189

5.1 Curves Defined by Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5.3 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

5.4 Conic Sections in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

6 Review Material 219

6.1 Midterm 1 Review Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.2 Midterm 2 Review Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

6.3 End of Term Review Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

6.4 Final Exam Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

6.5 Final Exam Practice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Bibliography 257

Index 258

Preface

This booklet contains our notes for courses Math 150/151 - Calculus I at Simon Fraser University. Stu-dents are expected to use this booklet during each lecture by follow along with the instructor, filling in thedetails in the blanks provided, during the lecture.

Definitions of terms are stated in orange boxes and theorems appear in blue boxes .

Next to some examples you’ll see [link to applet]. The link will take you to an online interactive applet toaccompany the example - just like the ones used by your instructor in the lecture. Clicking the link abovewill take you to the following website containing all the applets:

http://www.sfu.ca/ jtmulhol/calculus-applets/html/appletsforcalculus.html

Try it now.

No project such as this can be free from errors and incompleteness. We will be grateful to everyonewho points out any typos, incorrect statements, or sends any other suggestion on how to improve thismanuscript.

Veselin JungicSimon Fraser [email protected]

Jamie MulhollandSimon Fraser Universityj [email protected]

August 8, 2018

iii

http://www.sfu.ca/~jtmulhol/calculus-applets/html/appletsforcalculus.html

Greek Alphabet

lowercase

capital name pronunciation lowercase

capital name pronunciation

α A alpha (al-fah) ν N nu (new)β B beta (bay-tah) ξ Ξ xi (zie)γ Γ gamma (gam-ah) o O omicron (om-e-cron)δ ∆ delta (del-ta) π Π pi (pie)ε E epsilon (ep-si-lon) ρ P rho (roe)ζ Z zeta (zay-tah) σ Σ sigma (sig-mah)η H eta (ay-tah) τ T tau (taw)θ Θ theta (thay-tah) υ Υ upsilon (up-si-lon)ι I iota (eye-o-tah) φ Φ phi (fie)κ K kappa (cap-pah) χ X chi (kie)λ Λ lambda (lamb-dah) ψ Ψ psi (si)µ M mu (mew) ω Ω omega (oh-may-gah)

v

Part 1

Review: Functions and Models

1

PART 1: FUNCTIONS AND MODELS LECTURE 1.0 BASIC SETS OF NUMBERS 2

Basic sets of numbers

• natural numbers: the set of counting numbers

N = {1, 2, 3, 4, 5, . . .}

(Some authors include 0 in this set.)

• integers: the set of natural numbers with their negatives

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}

• rational numbers: the set of ratios of integers

Q ={ a

b: a, b ∈ Z, b 6= 0

}

• real numbers R:These are more difficult to define, but we already have an intuitive idea of what they are.

They include all the rational numbers Q and all the numbers which fill in all the gaps between therational numbers.

PART 1: FUNCTIONS AND MODELS LECTURE 1.1 FOUR WAYS TO DEFINE A FUNCTION 3

1.1 Four Ways to Define a Function

1. Definition. A function (or map) is a rule or correspondence that associates each element of a setX, called the domain, with a unique element of a set Y , called the codomain.

2. The range of f is the set of all elements in Y which correspond to an element of X:

range f = {f(x) : x ∈ X}.

3. Example. The following function maps each person to their age.

domain = codomain =

range =


4. Reminder. In calculus we will only consider functions whose domain and codomain consist of real numbers.Functions can then be described in various ways:

(a) verbally (word description)

ex. The area of a circle is π times the radius squared.

(b) algebraically (by a formula)

ex. A(r) = πr2

(c) numerically (by a table of values)ex.

time (s) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6velocity (m/s) 0 0.2 0.5 0.8 1.0 0.6 0.2 0 -0.1

or by a set of ordered pairs

{(0, 0), (0.2, 0.2), (0.4, 0.5), (0.6, 0.8), (0.8, 1.0), (1.0, 0.6), (1.2, 0.2), (1.4, 0), (1.6,−0.1)}

(d) visually (by a graph)

5. Example. Let f(x) = x2.

(a) Find the following values: f(2), f(−1), f(0), f(2/3), f(√

2), f(π), f(a+ h).


(b) Sketch the graph of f .

6. A 10-ft wall stands 5 ft from a building and a ladder of variablelength L, supported by the wall, is placed so it reaches from theground to the building. Let y denote the vertical distance fromthe ground to where the tip of the ladder touches the building,and let x denote the horizontal distance from the wall to thebase of the ladder.

(a) Find an expression for the height y as a function of x.(b) Find an expression for the length L as a function of x.(c) Determine the domain and range of the function L(x)

found in part (b).


7. Reminder. If a function is given by a formula and the domain is not stated explicitly, the conventionis that the domain is the set of all numbers for which the formula makes sense and defines a realnumber.

(a) Find the domain of the functiong(x) =

1

x2 − x.

(b) Find the domain of the functionh(t) =

√16− t2.

What is the range?

8. Reminder. The graph of a function f is defined to be the set of all points (x, y) in the Cartesianplane satisfying the equation

y = f(x).

Sketch the graphs of the following functions.

(a) f(x) = x+ 1

(b) g(t) = t2 + 1


(c) h(x) =

{2x+ 3 if x ≤ 0x2 + 3 if x > 0

(d) f(x) = |x|

9. Reminder. Vertical line test for testing whether a curve is the graph of a function.If every vertical line intersects the curve at most once then the curve is a graph of a function.

Which curve is the graph of a function?


10. Additional Notes

PART 1: FUNCTIONS AND MODELS LECTURE 1.2 CATALOG OF ESSENTIAL FUNCTIONS 9

1.2 Mathematical Models: A Catalog of Essential Functions

1. Lines (linear function):A line is determined by two bits of information: and .

Or, equivalently, by .

2. Find the equation of the line in each of the following cases.

(a) slope = 2, containing P = (1, 3).

(b) containing the points (1, 3) and (−2, 7).


3. Some Common Functions:

(a) Power Functions: A power function is a function of the form

f(x) = xa

where a is a fixed real number.


(b) Polynomials: A polynomial is a function of the form

f(x) = anxn + an−1x

n−1 + . . .+ a2x2 + a1x+ a0

where n is an integer and the ai are fixed real numbers, which are called the coefficients of f .


(c) Rational Functions: A rational function is the ratio of two polynomials:

f(x) =p(x)

q(x)

where p(x) and q(x) are polynomials.


4. Trigonometric Functions: Let us recall the trigonometric functions sine, cosine, and tangent.

Why are there 360◦ in a full rotation? (◦ is read ”degrees”)

Radian measure of an angle:

5. Sketch the following angles (radians) in standard position and give the measure of the angle indegrees:

a)π

2b)π

4c) −5π

6d)

13π

3


6. Determine the coordinates of the point where the terminal side of the angle intersects the unit circle.

a)π

4b)π

3

7. Definition: The sin and cos of an angle:


8. We can fill out the following table

θ sin θ cos θ tan θ

0 0 1 0

π

6(30◦)

1

2

√3

2

1√3

π

4(45◦)

1√2

1√2

1

π

3(60◦)

√3

2

1

2

√3

π

2(90◦) 1 0 −

9. Sketching the graphs of sin and cos.

PART 1: FUNCTIONS AND MODELS LECTURE 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS 18

1.3 New Functions From Old Functions

1. Sketch the graphs of y = x2 + 2, y = (x− 1)2, and y = (x− 1)2 + 2.

2. Sketch the graph of y = 3x2 − 6x+ 1.


3. Vertical & Horizontal Shifts:Suppose c > 0. To obtain the graph ofy = f(x) + c, shift the graph of y = f(x) a distance c units upwardy = f(x)− c, shift the graph of y = f(x) a distance c units downwardy = f(x− c), shift the graph of y = f(x) a distance c units to the righty = f(x+ c), shift the graph of y = f(x) a distance c units to the left

4. Sketch the graph of y = sin 2x.

5. Vertical and Horizontal Streching and ReflectingSuppose c > 1. To obtain the graph ofy = cf(x), stretch the graph of y = f(x) vertically by a factor of cy = 1cf(x), compress the graph of y = f(x) vertically by a factor of cy = f(cx), compress the graph of y = f(x) horizontally by a factor of cy = f(x/c), stretch the graph of y = f(x) horizontally by a factor of cy = −f(x), reflect the graph of y = f(x) about the x-axisy = f(−x), reflect the graph of y = f(x) about the y-axis

6. Given the graph of f sketch the graph of 12 (f(x− 4)− 8).


7. Algebra of FunctionsWe can combine functions in different ways to create new functions.Let f and g be two functions. The sum f+g, the difference f−g, the product fg, and the quotientf

gare defined as follows:

(f + g)(x) = f(x) + g(x) domain = A ∩B(f − g)(x) = f(x)− g(x) domain = A ∩B(fg)(x) = f(x) · g(x) domain = A ∩B(f

g

)(x) =

f(x)

g(x)domain = {x ∈ A ∩B | g(x) 6= 0}

8. Example. If f(−1) = 1, f(2) = 3, g(−1) = −5 and g(2) = 17 find (f + g)(−1), (fg)(−1) and (f/g)(2).

9. Composition of FunctionsAnother way we can define new functions from old ones is by composition. If f and g are two functionswe write

(f ◦ g)(x) = f(g(x))

for the function obtained by applying f to the output of g. The function f ◦g is called the compostionof f with g.The domain of the composite function f ◦ g is the set of all x such that

(a) x is in the domain of g,(b) g(x) is in the domain of f .


10. Example. If f(x) = x2 and g(x) = 2x+ 1 find f ◦ g and g ◦ f .

11. Example. If f(x) = 2x, g(t) = 3√t and h(θ) = sin θ find h ◦ f ◦ g. What is the domain of h ◦ f ◦ g?

12. Example. Given F (x) = (1 + 2 sinx)3 find functions f , g and h such that F = f ◦ g ◦ h.

PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 23

1.4 Exponential Functions & Inverse Functions and Logarithms

(This lecture corresponds to Sections 1.4 and 1.5 of Stewart’s Calculus.)

1. Reminder. For all a ∈ (0, 1) ∪ (1,∞) and all x, y ∈ R:

(a) ax+y = ax · ay

(b) ax−y =ax

ay

(c) (ax)y = axy

(d) (ab)x = ax · bx

2. Reminder. Sketch the graphs of the functions f(x) = 2x and g(x) = 3x.

3. Reminder. Sketch the graph of the function F (x) =(

1

2

)x.


4. Reminder. Evaluate f(2), f(−2), f( 12 ) and f(32 ) if f(x) = 4

x.

5. BIG Question. What is 4√2?


6. Reminder. Napier’s constant:

e ≈ 2.718281828459045235360287471352

(John Napier, 1550-1617)


7. Definition. A function f is called a one-to-one function if it never takes on the same value twice;that is

if x1 6= x2 then f(x1) 6= f(x2) .

8. Example. Which of the following functions are one-to-one?

(a) f(x) = x2

(b) g(x) = x3

(c) h(x) = ex

(d) i(x) = sinx(e) j(x) = sinx, x ∈

[−π2 ,

π2

]


9. Horizontal Line Test. A function is one-to-one if and only if no horizontal line intersects its graphmore than once.

10. Definition. Let f be one-to-one function with domain A and range B. Then its inverse functionhas domain B and range A and is defined by

f−1(y) = x ⇔ f(x) = y

for any y ∈ B.


11. Example. Find a formula for the inverse of f(x) = x3x+ 1

.

12. Logarithmic Function. The inverse function of the exponential function f(x) = ax is called thelogarithmic function with base a.

13. All You Need To Know. For any a > 0, a 6= 1, any x > 0, and any y ∈ R

loga x = y ⇔ ay = x

14. Example. Determine log2 (16), log2 ( 18 ) and log2 (1).

15. Example. Can you find log2 (−32)?

16. Reminder. For all a ∈ (0, 1) ∪ (1,∞) and any positive x and y:

(a) loga(xy) = loga x+ loga y

(b) loga(xy

)= loga x− loga y

(c) loga(xr) = r loga x (r is a real number)

17. Notation.log10 x = log x

loge x = lnx


18. Reminder. Sketch the graph of the function y = lnx

19. Example. Solve the equation ex3−3 − 9 = 0 for x.

20. Inverse Trig Functions. Here we will limit our discussion to sin.

21. Definition. The inverse function of the sine function f(x) = sinx, −π2 ≤ x ≤π2 , is called arcsine

and is denoted by either sin−1 or arcsin.


22. All You Need To Know. For any −1 ≤ x ≤ 1, and any −π2 ≤ y ≤π2

sin−1 x = y ⇔ sin y = x

23. Determine the following.

(a) sin−1(√

32 )

(b) sin(sin−1( 13 ))

(c) sin−1(sin( 3π4 ))

PART 1: FUNCTIONS AND MODELS LECTURE R PREPARATION FOR LIMITS 32

Review: Preparation for Limits

(This lecture corresponds to Chapter 1 of the Companion Guide available on Canvas website.)

1. Algebraic Simplification of Functions: Give the domain and a simplified expression for

f(x) =x2 − 4x− 2

.

Sketch the graph of f .

2. Consider the function g(x) = x+ 2. Does f(x) = g(x)?


3. Factor Theorem. Let P (x) be a polynomial and r a real number. If P (r) = 0, then x−r is a factor ofP (x), i.e. P (x) = (x−r)Q(x) for some polynomial Q(x). Also, if x−r is a factor of P (x), then P (r) = 0.

4. Example. Factor x3 − 3x2 − 13x+ 15.

5. Example. Consider the rational function

f(x) =x3 + 5x2 + 5x+ 4

x2 + x− 12.

Give the domain of f . Find any common factors of the numerator and denominator, and write f(x) ina simplified form.


6. Quotients with Radicals: Give the domain and a simplified expression for

m(h) =

√9 + h− 3

h.

7. Quotients with Absolute Values: Give the domain and a simplified expression for

h(x) =|3− 6x|2x− 1

.

Sketch the graph of h.


8. Additional Notes

PART 2: LIMITS AND DERIVATIVES LECTURE R PREPARATION FOR LIMITS 36

Part 2

Limits and Derivatives

37

PART 2: LIMITS AND DERIVATIVES LECTURE 2.1 TANGENT AND VELOCITY PROBLEMS 38

2.1 The Tangent and Velocity Problems

(This lecture corresponds to Section 2.1 of Stewart’s Calculus.)

1. Quote. If I were again beginning my studies, I would follow the advice of Plato and start with math-ematics.Galileo Galilei, Italian philosopher and astronomer, 1564-1642.

2. The Tangent Problem. Find an equation of the tangent line ` to a curve with equation y = f(x) ata given point P .

3. Three Questions.

(a) What is the tangent line ` to a curve with equation y = f(x) at a given point P ?

(b) If a curve with equation y = f(x) and a point P on the curve are given, does the tangent `exist?

(c) If a curve with equation y = f(x) and a point P = (x0, f(x0)) are given and if the tangent line `exists then an equation of ` is given by

y − f(x0) = m(x− x0) .

How do we calculate the slope m?

4. Hint. Find the slopes of the secant lines to the parabola y = x2 through the points (1, 1) and:

(a) (2, 4)


(b) (1.5, 1.52)

(c) (1.1, 1.12)

(d) (1.001, 1.0012)

5. BIG Question. What if the second point is VERY, VERY close to the point (1, 1)?

6. Velocity Problem. By definition

avarage velocity =distance traveled

time elapsed

What if the period of time elapsed is very small?

7. Example. The position of the car is given by the values in the table.

t 0 1 2 3 4 5s 0 10 32 70 119 178

where t is in seconds and s is in feet.

Find the average velocity for the time beginning when t = 2 and lasting

(a) 3 seconds


(b) 2 seconds

(c) 1 second

8. Question. What is the meaning of the number that we see on the car speedometer as we travel incity traffic?

9. Answer. The number represents the instantaneous velocity.

PART 2: LIMITS AND DERIVATIVES LECTURE 2.2 THE LIMIT OF A FUNCTION 42

2.2 The Limit of a Function


1. Quote. “Black holes are where God divided by zero.”Steven Wright, American comedian, 1955-

2. Problem. Let f(x) = x2 − x− 2x− 2

.

(a) Determine the domain of f .

(b) Complete the tablex f(x) x f(x)1 31.9 2.11.99 2.011.999 2.0011.9999 2.0001


3. Definition. We writelimx→a

f(x) = L

and say

”the limit of f(x), as x approaches a, equals L”

if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to besufficiently close to a (on either side of a) but not equal to a.

4. Example. Guess the value oflimx→0

sinx

x.

5. Problem. What can we say about

limx→0

|x|x

?

6. Definition. We writelimx→a+

f(x) = L

and say

”the right-hand limit of f(x), as x approaches a, equals L”

if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to besufficiently close to a and x greater than a.


7. Example. Sketch the graph of the function

f(x) =

x+ 1 if x ≤ −1x2 if x ∈ (−1, 0)1 if x = 0x2 if x ∈ (0, 1]x+ 1 if x > 1

Find

(a) limx→−1−

f(x)

(b) limx→−1+

f(x)

(c) limx→0−

f(x)

(d) limx→0+

f(x)

(e) limx→0

f(x)

8. Fact.limx→a

f(x) = L ⇐⇒ ( limx→a−

f(x) = L and limx→a+

f(x) = L)


9. Problem. Sketch the graph of f(x) = 1(x+ 1)2

.

10. Definition. Let f be a function defined on both sides of a, except possibly at a itself. Then

limx→a

f(x) =∞

means that the values of f(x) can be made arbitrarily large (as large as we please) by taking xsufficiently close to a, but not equal to a.

11. Examples. Sketch the graph of the following function.

g(x) =x+ 3

x− 1

12. Read Example 10 in text regarding f(x) = tan (x)

13. Definition. The line x = a is called a vertical asymptote of the curve y = f(x) if at least one ofthe following statements is true:

limx→a

f(x) =∞ limx→a−

f(x) =∞ limx→a+

f(x) =∞limx→a

f(x) = −∞ limx→a−

f(x) = −∞ limx→a+

f(x) = −∞

PART 2: LIMITS AND DERIVATIVES LECTURE 2.3 CALCULATING LIMITS 47

2.3 Calculating Limits Using the Limit Laws


1. Quote. “Laws are like sausages. It’s better not to see them being made.”Otto von Bismarck , German statesman, 1815 - 1898)

2. Example. Guess the value of limt→0

√t+ 9− 3

t.

3. Limit Laws. Suppose that c is a constant and the limits

limx→a

f(x) and limx→a

g(x)

exist. Then

(a) limx→a

(f(x) + g(x)) = limx→a

f(x) + limx→a

g(x)

(b) limx→a

(f(x)− g(x)) = limx→a

f(x)− limx→a

g(x)

(c) limx→a

(c · f(x)) = c · limx→a

f(x)

(d) limx→a

(f(x) · g(x)) = limx→a

f(x) · limx→a

g(x)

(e) limx→a

f(x)

g(x)=

limx→a f(x)

limx→a g(x)if limx→a g(x) 6= 0.

(f) limx→a

[f(x)]p/q = [ limx→a

f(x)]p/q

4. Two Special Limit Laws.

(a) limx→a

c = c

(b) limx→a

x = a


5. Example. Evaluate limx→2

(x3 + 3x2 − 4x+ 5).

6. Direct Substitution Property. If f is a polynomial or a rational function and a is in the domain off , then

limx→a

f(x) = f(a) .

7. Examples. Find the following limits.

(a) limx→−1

x+ 1

x3 + 1

(b) limt→0

√t+ 9− 3

t

(c) limh→0

f(x+ h)− f(x)h

if f(x) = x2


8. Example. Find limx→0

x2

|x|Reminder.

limx→a

f(x) = L ⇐⇒ ( limx→a−

f(x) = L and limx→a+

f(x) = L)

9. Theorem. If f(x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both existas x approaches a, then

limx→a

f(x) ≤ limx→a

g(x) .

10. Squeeze Theorem. If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and

limx→a

f(x) = limx→a

h(x) = L

thenlimx→a

g(x) = L .


11. Example. Show that

limx→0

[x2(

sin1

x+ cos

1

x

)]= 0 .

PART 2: LIMITS AND DERIVATIVES LECTURE 2.4 THE PRECISE DEFINITION OF LIMIT 52

2.4 The Precise Definition of Limit


1. Quote.

“There’s a delta for every epsilon,It’s a fact that you can always count upon.

There’s a delta for every epsilonAnd now and again,There’s also an N.”

(Tom Lehrer, American singer-songwriter, satirist, pianist, and mathematician, 1928 - .)

2. The �, δ Game. Consider the function f(x) = 3x − 1 and the point x = 1. There are two players inthis game: Player A and Player B. The game is played as follows. Player A chooses a number, say �.The object of Player B is to find a number δ so that all values in the interval (1− δ, 1 + δ) have imagein the interval (f(1)− �, f(1) + �). The winner is determined as follows:

1) If Player A can pick a number � such that Player B cannot find such a δ then Player A wins.2) If Player B can find a δ for any � given by Player A then Player B wins.

Who wins Player A or Player B?


3. Definition. Let f be a function defined on some open interval that contains the number a, exceptpossibly at a iself. Then we say that the limit of f(x) as x approaches a is L, and we write

limx→a

f(x) = L

if for every number ε > 0 there is a δ > 0 such that

|f(x)− L| < ε whenever 0 < |x− a| < δ .

[link to applet]

http://www.geogebratube.org/student/m47174


4. Example. Prove the statement using the �, δ definition.

limx→3

(2− 5x) = −13


5. To Be Continued ... ... in Math 242.

PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 56

2.5 Continuity


1. Quote. “If I were asked to name, in one word, the pole star round which the mathematical firmamentrevolves, the central idea which pervades the whole corpus of mathematical doctrine, I should pointto Continuity as contained in our notions of space, and say, it is this, it is this! ”(JJ Sylvester, English mathematician, 1814-1897)

2. Example. What is the difference between the two graphs?

3. Definition. A function f is continuous at a number a if

limx→a

f(x) = f(a) .

4. Note.

(a) a belongs to the domain of f(b) lim

x→af(x) exists

(c) limx→a

f(x) = f(a)


5. Definition. If(1) f is defined on an open interval containing a, except perhaps at a, and(2) f is not continuous at a

we say that f is discontinuous at a.

6. Example. Where are each of the following functions discontinuous?

(a)

f(x) =

x2 − 4x− 2

if x 6= 25 if x = 2

(b)

g(x) =

{ 1x− 2

if x 6= 25 if x = 2

(c)

h(x) =

{1 if x ∈ [1, 2)2 if x ∈ [2, 3)


7. Definition. A function f is continuous from the right at a number a if

limx→a+

f(x) = f(a)

and f is continuous from the left at a if

limx→a−

f(x) = f(a) .

8. Definiton. A function f is continuous on an interval if it is continuous at every number in thatinterval. We understand continuous at the endpoint to mean continuous from the right or continuousfrom the left.

9. Example. Find the number c that makes f(x) continuous for every x.

f(x) =

x4 − 1x3 − 1

if x 6= 1

c if x = 1


10. Fact. The following types of functions are continuous on their domains:

(a) polynomials(b) rational functions(c) root functions(d) trigonometric functions(e) inverse trigonometric functions(f) exponential functions(g) logarithmic functions

11. More Facts. If f and g are continuous at a and c is a constant, then the following functions are alsocontinuous at a:

f + g, f − g, cf, fg, fg

if g(a) 6= 0 .

12. Example. For which a, b ∈ R is the function

f(x) =

√1− x− 1ax

if x ∈ (0, 1]

1 if x = 0

bx4 + bx

x2 + xif x ∈ (−1, 0)

continuous on (−1, 1]?


13. Theorem. If f is continuous at b and limx→a

g(x) = b then

limx→a

f(g(x)) = f( limx→a

g(x)) = f(b) .

14. Example. Evaluate

limx→0

e√

1−x−1x .

15. Theorem. If g is continuous at a and f is continuous at g(a), then the composite function f ◦ g givenby (f ◦ g)(x) = f(g(x)) is continuous at a.


16. Intermediate Value Theorem. Suppose that f is continuous on the closed interval [a, b] and let Nbe any number between f(a) and f(b), where f(a) 6= f(b). Then there exists a number c in (a, b) suchthat f(c) = N .

17. Example. Use the Intermediate Value Theorem to prove that√

2 exists, i.e., prove that there is c ∈ Rsuch that c2 = 2.


18. Example. Use the Intermediate Value Theorem to show that the equation

ex = 2− x

has at least one real solution.

PART 2: LIMITS AND DERIVATIVES LECTURE 2.6 LIMITS AT INFINITY 64

2.6 Limits at Infinity: Horizontal Asymptotes


1. Quote. “Infinity is a floorless room without walls or ceiling. ”

(Anonymous)

2. Definition. Let f be a function defined on some interval (a,∞). Then

limx→∞

f(x) = L

means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large.

3. Problem. Sketch the graphs of the following functions

(a) f(x) =1

x

(b) g(x) = ex

(c) h(x) = tan−1 x

(d) i(x) =1

1 + x2


4. Definition. The line y = L is called a horizontal asymptote of the curve y = f(x) if either

limx→∞

f(x) = L or limx→−∞

f(x) = L .

5. Fact. If r > 0 is a rational number, then

limx→∞

1

xr= 0 .

If r > 0 is a rational number such that xr is defined for all x then

limx→−∞

1

xr= 0 .

6. Example. Evaluate

(a)

limx→∞

3x3 − 4x2 − 16x3 + x+ 2

(b)

limx→−∞

√3x2 − 5− 1

2x+ 5


(c)lim

x→−∞(√x2 + ax−

√x2 + bx)

7. Problem. Find the following limits.

(a) limx→∞

x2

(b) limx→∞

x2 + 2x− 1x3 + 3

(c) limx→∞

x4 + 5x3 − 1x2 + x+ 1

(d) limx→∞

ex

(e) limx→∞

ex

x2


8. Additional Notes

PART 2: LIMITS AND DERIVATIVES LECTURE R PROBLEM SOLVING, RATES OF CHANGE 68

Review: Problem Solving and Rates of Change


1. Quote. ”If you can’t solve a problem, then there is an easier problem you can solve: find it.”(George Polya , Hungarian Mathematician, 1887- 1985)

2. Polya’s Approach to Problem Solving:Polya developed a four-step approach to problem solving. The steps are:

(a) Define the problem.(b) Devise a plan for solving the problem.(c) Carry out the plan.(d) Test and evaluate the results.

3. A 10-ft wall stands 5 ft from a building. Suppose a ladder is placed on the ground so that it leansagainst the building and touches the top of the 10-ft wall as indicated in the picture. How much doesthe length of the ladder need to be adjusted by if the distance from the base of the ladder to the wallincreases from 5 ft to 7 ft?


4. Average Rate of Change.The average rate of change of a function f over the interval [x1, x2] is defined as

f(x2)− f(x1)x2 − x1

.

Notice that this can be interpreted as the slope of the secant line through the points (x1, f(x1)) and(x2, f(x2)).

5. Example. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottomof the tank in an hour, then Torricelli’s Law gives the volume V of water remaining in the tank aftert minutes as

V = 100, 000

(1− t

60

)20 ≤ t ≤ 60 .

Find the average rate at which the water is flowing out of the tank between:

(a) 0 and 10 minutes,(b) 40 and 50 minutes.

What is the average rate at which the water is flowing out of the tank over the entire 60 minute timeperiod?


6. Additional Notes

PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 71

2.7 Derivatives and Rates of Change


1. Quote. ”The real voyage of discovery consists not in seeking new landscapes, but in having new eyes.”(Marcel Proust, French author, 1871- 1922)

2. Definition. The tangent line to the curve y = f(x) at the point P (a, f(a)) is the line through Pwith slope

m = limx→a

f(x)− f(a)x− a

provided that this limit exists.

3. Note. If limx→a

f(x)− f(a)x− a

exists then

limx→a

f(x)− f(a)x− a

= limh→0

f(a+ h)− f(a)h


4. Example.

(a) Find the slope of the tangent line to the graph of f(x) = x3 at the pointi. x = 1

ii. x = 2

(b) Find the equation of the tangent line at each of the points above.

5. Example.

(a) Find the slope of the tangent to the curve

y =1√x

at the point where x = a.


(b) Find the equation of the tangent line at the point (1, 1).

6. The Most Important Definition in This Course.

Definition of Derivative. The derivative of a function f at a number a, denoted by f ′(a), is

f ′(a) = limh→0

f(a+ h)− f(a)h

if this limit exists.

7. Note. If limx→a

f(x)− f(a)x− a

exists then f ′(a) = limx→a

f(x)− f(a)x− a

= limh→0

f(a+ h)− f(a)h

8. Example. Find the derivative of the function

y =1

x− 1

at the point where x = 3.

9. Example. The following limit represents the derivative of some function f at some number a. Statef and a.

limh→0

2h+3 − 8h


10. Example. Let f(x) = |x|. Does f ′(0) exist?

11. Must Know! An equation of the tangent line to y = f(x) at (a, f(a)) is given by

y − f(a) = f ′(a)(x− a) .

12. Example. Find the equation of the tangent line to f(x) = 1x− 1

at the point where x = 3.

13. Compare the derivatives at each of the points on the graph.


14. Reminder. By definition

average velocity =displacement

time

15. More Precisely... Suppose an object moves along a straight line according to an equation of motions = f(t), where s is the displacement of the object from the origin at time t.

The average velocity of the object in the time interval from t = a to t = a+ h is given by

average velocity =f(a+ h)− f(a)

h.

16. BIG Question. What if h is small?17. Definition. We define the velocity (or instantaneous velocity) v(a) at time t = a as

v(a) = limh→0

f(a+ h)− f(a)h

.

18. Example. If an arrow is shot upward on the moon with a velocity of 58 m/s, its height (in meters)after t seconds is given by

H = 58t− 0.83t2 .

(a) Find the velocity of the arrow when t = a.


(b) When will the arrow hit the moon?

(c) With what velocity will the arrow hit the moon?

19. Rates of Change. Let f be a function defined on an interval I and let x1, x2 ∈ I. Then the incre-ment of x is defined as

∆x = x2 − x1and the corresponding change in y is

∆y = f(x2)− f(x1) .

The average rate of change of y with respect to x over the interval [x1, x2] is defined as

∆y

∆x=f(x2)− f(x1)

x2 − x1.

20. Must Know! The instantaneous rate of change of y with respect to x is defined as

lim∆x→0

∆y

∆x= limx2→x1

f(x2)− f(x1)x2 − x1

.


21. Example. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottomof the tank in an hour, then Torricelli’s Law gives the volume V of water remaining in the tank ingallons after t minutes as

V = 100, 000

(1− t

60

)20 ≤ t ≤ 60 .

Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of Vwith respect to t) as a function of t. What are the units?


22. Example. The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at aprice of p dollars per pound is Q = f(p).

(a) What is the meaning of the derivative f ′(8)? What are the units?

(b) Is f ′(8) positive or negative? Explain.

PART 2: LIMITS AND DERIVATIVES LECTURE 2.8 THE DERIVATIVE AS A FUNCTION 80

2.8 The Derivative as a Function


1. Quote. “I turn away with fear and horror from this lamentable sore of continuous functions withoutderivatives.”(Charles Hermite, French mathematician, 1822-1901.)

2. Reminder. The derivative of a function f at a number a, denoted by f ′(a), is

f ′(a) = limh→0

f(a+ h)− f(a)h

if this limit exists.

3. Find the derivative of the function f(x) = x2 at

(i) x = 0,(ii) x = 1,

(iii) x = 2,(iv) x = 10.


4. Problem. If a function f : I → R is given, find the set J ⊂ I such that f ′(x) exists for each x ∈ J .If J 6= ∅ then this new function f ′ : J → R is called the derivative of f .

5. Example. Letf(x) = x

23 =

3√x2 .

(i) Determine the domain of f .

(ii) Determine the formula for f ′(x). What is the domain of f ′?


(iii) Sketch graphs of f and f ′.


6. The graph of f is given. Sketch the graph of f ′.

7. Notation. For y = f(x) it is common to write:

f ′(x) = y′ =dy

dx=df

dx=

d

dxf(x) = Df(x) = Dxf(x)

Also,

f ′(a) =dy

dx

∣∣∣∣x=a

=dy

dx

]x=a

.

8. Definition. A function is differentiable at a if f ′(a) exists. It is differentiable on an openinterval (a, b) [or (a,∞) or (−∞, a) or (−∞,∞)] if it is differentiable at every number in the interval.


9. Two Questions.

(i) Is every continuous function differentiable?

(ii) Is every differentiable function continuous?

10. Three Cases. A function f is not differentiable at a number a from its domain if:

(i) The graph of f has a corner at the point (a, f(a));

(ii) f is not continuous at a;

(iii) The graph of f has a vertical tangent line when x = a.


11. Higher Derivatives. Suppose that f is a differentiable function. The second derivative of f isthe derivative of f ′.Notation.

(f ′)′ = f ′′

(y′)′ = y′′

d

dx

(dy

dx

)=d2y

dx2

12. Example. Find f ′′(x) if f(x) = x2.

13. Acceleration. The instantaneous rate of change of velocity with respect to time is called theacceleration of the object.

a(t) = v′(t) = s′′(t).

14. Example. The figure shows the graphs of three functions. One is the position function of a particle,one is its velocity, and one is its acceleration. Identify each curve.

Part 3

Differentiation Rules

87

PART 3: DIFFERENTIATION RULES LECTURE 3.1 DERIVATIVES: POLYNOMIALS AND EXP 88

3.1 Derivatives of Polynomials and Exponential Functions


1. Quote. “Young man, in mathematics you don’t understand things, you just get used to them.” (Johnvon Neumann, Hungarian mathematician and polymath, 1903-1957)

2. Reminder. The derivative of a function f is the function f ′ defined by

f ′(x) = limh→0

f(x+ h)− f(x)h

for all x for which this limit exists. Recall that we also use the notation ddx (f(x)) = f′(x) for the

derivative.

3. Must Know!

(a) Derivative of a Constant.d

dx(c) = 0

(b) We have already seen that the following are true:

d

dx(x) = 1,

d

dx(x2) = 2x,

d

dx(x3) = 3x2.

You may be able to see a pattern. In fact, we have the following rule.

The Power Rule. If n is any real number, then

d

dx(xn) = nxn−1


(c) Constant Multiple Rule. If c is a constant and f is a differentiable function, thend

dx(cf(x)) = c · d

dxf(x)

(d) Sum Rule. If f and g are differentiable functions, thend

dx(f(x) + g(x)) =

d

dxf(x) +

d

dxg(x)

(e) The Derivative of a Polynomial. If

p(x) = anxn + an−1x

n−1 + . . .+ a1x+ a0

where n is a nonnegative integer and an 6= 0 then

p′(x) = nanxn−1 + (n− 1)an−1xn−2 + . . .+ a1 .


4. Example. Find an equation of the tangent line to the curve y = 2x3 − 7x2 + 3x+ 4 at the point (1, 2).

5. Example. Find an equation for the straight line that passes through the point (0, 2) and it is tangentto the curve y = x3.

6. Fact. If f(x) = ax, a > 0, a 6= 1, is an exponential function then

f ′(0) = limh→0

ah − 1h

exists.

7. Fact It is straightforward to show that if f(x) = ax then

f ′(x) = f ′(0) · ax .


8. Must Know! e is is the number such that

limh→0

eh − 1h

= 1 .

e ≈ 2.71828

9. Derivative of the Natural Exponential Function. If f(x) = ex is the natural exponential functionthen

f ′(x) = f(x) .

Thusd

dx(ex) = ex .

10. Example. Differentiate the function

f(x) = 2x3 + 3x23 − ex+2 .


11. Example. At what point on the curve y = ex is the tangent line parallel to the line y = 2x?

PART 3: DIFFERENTIATION RULES LECTURE 3.2 THE PRODUCT AND QUOTIENT RULES 94

3.2 The Product and Quotient Rules


1. Quote. ”Five out of four people have trouble with fractions.”

(Steven Wright, American comedian, 1955-)

2. Problem. Suppose we have two functions f(x) = 3√x2 and g(x) = ex and we want to compute the

derivative of their product

d

dx(

3√x2ex).

How do we do this?

3. Product Rule. If f and g are both differentiable, then

d

dx[f(x)g(x)] = f(x)

d

dx[g(x)] + g(x)

d

dx[f(x)] .

In Lagrange’s notation this is written as (fg)′ = f · g′ + g · f ′.

4. Examples.

(a) Differentiate f(x) = 3√x2 · ex.


(b) Differentiate g(x) = (x+ 1)(2x2 − x+ 1).

5. Quotient Rule. If f and g are differentiable, then

d

dx

[f(x)

g(x)

]=g(x)

d

dx[f(x)]− f(x) d

dx[g(x)]

[g(x)]2.

In Lagrange’s notation this is written as(f

g

)′=g · f ′ − f · g′

g2.

6. Examples.

(a) Differentiate y =2t2 − 1t3 + 1

.

(b) Differentiate f(x) = e−x.


(c) If f(3) = 4, g(3) = 2, f ′(3) = −6, and g′(3) = 5, find the following numbers.i. (f + g)′(3)

ii. (fg)′(3)

iii.(f

g

)′(3)

iv.(

f

f − g

)′(3)


7. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.3 DERIVATIVES OF TRIG FUNCTIONS 98

3.3 Derivatives of Trigonometric Functions


1. Quote. ”Trigonometry is the mathematics of sound and music.”(Frank Wattenberg, American mathematician, 1952-)

2. Problem. What is the derivative of sinx?

3. Must Know!

(a)d

dx(sinx) = cosx

(b)d

dx(cosx) = − sinx

(c)d

dx(tanx) = sec2 x

(d)d

dx(secx) = secx tanx

(e)d

dx(cscx) = − cscx cotx

(f)d

dx(cotx) = − csc2 x


4. Problem. Prove thatd

dx(sinx) = cosx .

5. Trigonometric Limits. Above we used the very important results

limθ→0

sin θ

θ= 1 and lim

θ→0

cos θ − 1θ

= 0.

We now prove these results.


6. Examples.

(a) Differentiate y =1 + tanx

x− cotx.

(b) Find the points on the curvey =

cosx

2 + sinx

at which the tangent is horizontal.

7. A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladderand the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of theladder slides away from the wall, how fast does x change with respect to θ when θ = π/3?


8. Examples. Evaluate

(a) limx→0

sin 2x

x

(b) limθ→0

sin 2θ

cos θ − 1


9. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.4 CHAIN RULE 103

3.4 Chain Rule


1. Puzzle. A duck before two ducks, a duck behind two ducks, and a duck in the middle. How manyducks are there?

2. Reminder. The composition of the functions f and g is defined by

(f ◦ g)(x) = f(g(x)) .

3. Example. Let f(u) = sinu and g(x) = 1 + x2. Find F = f ◦ g.

4. Chain Rule. If f and g are both differentiable and F = f ◦ g is the composite function defined byF (x) = f(g(x)), then F is differentiable and F ′ is given by

F ′(x) = f ′(g(x)) · g′(x) .

In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then

dy

dx=dy

du· dudx

.

5. Examples.

(a) Let f(u) = sinu and g(x) = 1 + x2 and let F = f ◦ g. Find the derivative of F .


(b) Find y′ =dy

dx. if

i. y = (2− 5x)3

ii. y = (x+ sinx)5(1 + ex)2

iii. Express the derivative dy/dx in terms of x if

y = u5 and u =(4x− 1)2

x.


6. Examples. Find f ′.

(a)f(x) =

√2 + 5x2

(b)f(x) = (tan (x2))3

(c)f(x) = ecos x

7. Must Know!d

dx(ax) = ax ln a .


8. Examples.

(a) A pebble dropped into a lake creates an expanding circular ripple. Suppose that the radius ofthe circle is increasing at the rate of 2 in/s. At what rate is its area increasing when its radius is10 in?

(b) Suppose that f(0) = 0 and f ′(0) = 1. Calculate the derivative of f(f(f(x))) at x = 0.

(c) Under certain circumstances a rumor spreads according to the equation

p(t) =1

1 + ae−kt

where p(t) is the proportion of the population that knows the rumor at time t and a and k arepositive constants.

i. Find limt→∞

p(t).

ii. Find the rate of spread of the rumor.


9. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.5 IMPLICIT DIFFERENTIATION 108

3.5 Implicit Differentiation


1. Dictionary. implicitadjectiveDefinition:1. implied: not stated, but understood in what is expressedAsking us when we would like to start was an implicit acceptance of our terms.2. absolute: not affected by any doubt or uncertaintyimplicit trust3. contained: present as a necessary part of somethingConfidentiality is implicit in the relationship between doctor and patient.

2. Problem. The curvex3 + y3 = 3xy

is called the folium of Descartes. Find the equation of the tangent line at the point(

32 ,

32

).

3. Implicitly Defined Function. An equation in two variables x and y may have one or more solutionsfor y in terms of x or for x in terms of y. These solutions are functions that are said to be implicitlydefined by the equation.

4. Example

(a) x2 + y2 = 1

(b) x3 + y3 = 3xy


5. Implicit Differentiation.

(a) Use the chain rule to differentiate both sides of the given equation, thinking of x as the indepen-dent variable.

(b) Solve the resulting equation for dydx .

6. Example. The curvex3 + y3 = 3xy

is called the folium of Descartes. Find the equation of the tangent line at the point(

32 ,

32

).

7. Example. Suppose that water is being emptied from a spherical tank of radius 10 ft. If the depth ofwater in the tank is 5 ft and is decreasing at the rate of 3 ft/sec, at what rate is the radius r of the topsurface of the water decreasing?


8. Differentiation of an Inverse Function. Suppose f is a one-to-one differentiable function and itsinverse function f−1 is also differentiable. Use implicit differentiation to show that

(f−1)′(x) =1

f ′(f−1(x))

provided that the denominator is not 0.

9. Must Know!

(a)d

dx(sin−1(x)) =

1√1− x2

(b)d

dx(cos−1(x)) = − 1√

1− x2

(c)d

dx(tan−1(x)) =

1

1 + x2

(d)d

dx(cot−1(x)) = − 1

1 + x2


10. Example. Determine the points on the circle

(x− 1)2 + (y − 2)2 = 4

where the tangent line is horizontal or vertical.

11. Example. For the curvex2 + y2 = 5

find y′′ by implicit differentiation.


12. Orthogonal Trajectories.

(a) Two curves are called orthogonal if at each point of intersection their tangent lines are perpen-dicular.

(b) Two families of curves are orthogonal trajectories of each other if every curve in one familyis orthogonal to every curve in the other family.

(c) Show that the given families of curves are orthogonal trajectories of each other:

x2 + y2 = ax and x2 + y2 = by .

PART 3: DIFFERENTIATION RULES LECTURE 3.6 DERIVATIVE: LOGARITHMS 114

3.6 Derivatives of Logarithmic Functions


1. Quote. “One real estate development company advertised that an investment with it would growlogarithmically.”(From Ed Barbeaus column, Fallacies, Flaws, and Flimflam, in College Math. Journal 36 (2005),394-396.)

2. Must Know!d

dx(loga x) =

1

x ln a

3. When a = e this becomesd

dx(lnx) =

1

x.

4. Examples. Differentiate

(a) y = log2(3x2 + ex)

(b) y = ln(x+√x2 − 1)


(c) y =√

lnx

(d) y = ln√x

(e) y = ln(

x2

(x+ 3)4

)

5. More Examples. Differentiate

(a)

y =4√x3 5√x3 + 1

(2x+ 1)3


(b)y = xx

2

(c)y = ln |x|

6. Must Know!limx→0

(1 + x)1x = e


7. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.7 RATES OF CHANGE IN SCIENCE 118

3.7 Rates of Change in the Natural and Social Sciences


1. Quote. ”If you want to see practical applied mathematics, read chemical engineering; if you want tosee theoretical applied mathematics, read electrical engineering.And if you want to read pure math, read economics.”(Unknown blogger.)

2. Example (Physics). The equation of motion for a particle is given by

s = 2t3 − 3t2 − 12t, t ≥ 0

where s is in meters and t is in seconds.

(a) Find the velocity and acceleration as functions of t.(b) The graph of s = s(t) is shown. Sketch the graphs of the velocity and acceleration functions for

0 ≤ t ≤ 4.(c) When is the particle speeding up? Slowing down?(d) What does the expression s′′′(t) = a′(t) represent?

[link to applet]



3. Exercise.Let v(t) be a function which gives the velocity of a particle at time t. Consider the speed functionw(t) = |v(t)|.

(a) The particle is speeding up when w′(t) > 0. Show that this is equivalent to the condition thatv(t) and a(t) have the same sign.

(b) Similarly, the particle is slowing down when w′(t) < 0. Show that this is equivalent to thecondition that v(t) and a(t) have opposite signs.

[Hint: Remove the absolute value signs by writing w(t) as a piecewise defined function. Then differ-entiate the piecewise function, paying careful attention to the conditions which define each piece.]


4. Example (Chemistry). If one molecule C is formed from one molecule of the reactant A and onemolecule of the reactant B, and the initial concentrations of A and B have a common value [A] =[B] = a moles/L then

[C] =a2kt

akt+ 1

where k is a constant.

(a) Find the rate of reaction at time t.(b) Show that if x = [C], then

dx

dt= k(a− x)2

(c) What happens to the concentration as t→∞?(d) What happens to the rate of reaction as t→∞?


5. Example (Economics). Suppose that the cost (in dollars) for a company to produce x pairs of a newline of jeans is

C(x) = 2000 + 3x+ 0.01x2 + 0.0002x3

(a) Find the marginal cost function.(b) Find C ′(100) and explain its meaning. What does it predict?(c) Compare C ′(100) with the cost of manufacturing the 101st pair of jeans.


6. Example. The height of a certain cylinder is always twice its radius r. If its radius is changing, showthat the rate of change of its volume with respect to r is equal to its surface area.


7. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.8 EXPONENTIAL GROWTH AND DECAY 124

3.8 Exponential Growth and Decay


1. Quote. “It’s the whole issue with exponential growth, it’s very slow in the beginning but over thelong term it gets ridiculous.”(Drew Curtis, Founder and chief administrator of Fark.com, 1973 - )

2. Exponential Growth and Decay: A quantity q is said to be growing (or decaying) exponentially if

q = Aekt

where A and k are constants.

3. Natural Growth Equation. The solution of the initial-value problem

dy

dt= ky, y(0) = y0

isy(t) = y0e

kt .


4. Example. Calculopolis had a population of 25000 in 1980 and the population of 30000 in 1990.What population can the Calculopolis planers expect in the year 2020 if the population grows at arate proportional to its size?


5. Radioactive Decay. Radioactive material is known to decay at a rate proportional to the amountpresent. This means, if N is the amount (mass) of radioactive material at time t then it must satisfythe model:

dN

dt= −kN, k > 0.

6. Example. It takes 8 days for 20% of a particular radioactive material to decay. How long does it takefor 100 grams of material to decay to 50 grams? 40 grams? 0 grams?

7. Remark. Usually k is specified in terms of the half-life of the isotope

τ =ln 2

k.

This is the time required for half of any given quantity to decay.


8. Newton’s Law of Cooling and Heating. If a warm object is put in cooler surroundings its tempera-ture will steadily decrease. A law of physics known as Newtons law of cooling says that the rate atwhich the object cools is proportional to the difference between its temperature and the surroundingtemperature. This law is modeled by the differential equation:

dT

dt= k(T −M)

where

• T (t) is the temperature of the object at time t• M is the temperature of the surroundings (ambient temperature - which is constant)• k a constant (called the cooling constant)

9. Example. When a cold drink is taken from a refrigerator, its temperature is 5◦C. After 25 minutesin a 20◦C room its temperature has increased to 10◦C.

(a) What is the temperature of the drink after 50 minutes?(b) When will its temperature be 15◦C?


10. See Stewart’s textbook on page 241 for an example involving compound interest.


PART 3: DIFFERENTIATION RULES LECTURE R PREPARATION FOR RELATED RATES 129

Review: Preparation for Related Rates


1. Quote. ”Oh, yes these problems can be nasty. Lots of students fear related rates problems. Why?Maybe because they are word problems, and students just don’t like word problems. Having to changeEnglish into mathematics intimidates many people. It’s as if when they hear it in words, the mathe-matical sides of their brain shut down.”Colin Adams, Mathematician, author of How to Ace Calculus: The Streetwise Guide

2. Setting Up Equations for a Related Rates Problem:A related rates problem is a problem that has an equation relating two or more things which changeover time, and we want to find the derivative of one of the functions at a particular time. We’ll beginwith looking at how to set up these problems, as this is usually the part students find to be the mostdifficult, even though, surprisingly, it involves no calculus.

3. Example. A 6-ft tall man walks away from a 15-ft lamppost and the man’s shadow is cast on theground.(a) Draw a picture to represent this situation, and identify and label all constants and variables.(b) Determine the relationship between the distance x from the man to the lamppost and length of

the man’s shadow.

4. Example. An airplane is flying at a constant altitude of 9km above the ground and is approachinga camera on the ground. Let θ be the angle of elevation above the ground at which the camera ispointed. Find a relationship between the horizontal distance from the plane to the camera and theangle of elevation of the camera θ.


5. Example. Suppose the wood nymphs and satyrs are having a hot party and the wine is flowingfreely from the bottom of a giant cone-shaped barrel which is 10-ft high and 6-ft in radius at the top.Determine the relationship between the depth and the volume of the wine in the tank.

6. Example. A baseball player runs from home plate towards first base. Determine the relationshipbetween the distance x from the player to first base and the distance y from the player to second base.

7. Example. A plane flying at a constant speed of 300km/h passes over a ground radar station at analtitude of 1km and climbs at an angle of 30◦. If P denotes the location of the plane when the planepasses directly over the radar station determine the relationship between the distance from the radarstation to the plane and the distance from the plane to point P .


8. Interpreting a Rate of Change:In a related rates problem, statements are made about rates of change of certain quantities. Let’slook at how to interpret such statements.

9. Example. For each of the following statements, draw a picture, assign a letter name to each quantitythat is changing over time, and interpret the statement about rate of change in terms of symbols.

(a) After fueling up at a gas station a car drives west at 50 km/h.

(b) Water is being pumped out of a rectangular tank at a rate of 2m3/min

(c) A baseball player is running from home plate to first base and when the player is exactly halfwayto first base the distance from the player to second base is changing at a rate of 8m/s.

(d) A plane is flying at a constant speed 300km/h and climbing from a point P at an angle of 30◦.


10. Solving for a Rate of Change:In solving a related rates problem, once a relationship is found between all quantities and their ratesof change (i.e. derivatives) then the unknown rate (derivative) can be found by substituting in all thegiven information.

11. Example. If x and y are both differentiable functions of the variable t, and are related by the givenequation, then find the requested rate of change under the given conditions.

(a) sinx+ cos y = 1. Determinedy

dtwhen x = π/6, y = π/3 and

dx

dt= 2.

(b) x3 + y2 − xy = 8. Determine dxdt

when y = 0 anddy

dt= 8.

PART 3: DIFFERENTIATION RULES LECTURE 3.9 RELATED RATES 133

3.9 Related rates


1. Quote. “If you want to increase your success rate, double your failure rate. ”(Thomas John Watson, Sr., Founder of IBM, 1874 - 1956.)

2. A spherical balloon is being inflated. The radius r of the balloon is increasing at the rate of 0.2 cm/swhen r = 5 cm. At what rate is the volume V of the balloon increasing at that moment?


3. The Method of Related RatesWhen two variables are related by an equation and both are functions of a third variable (such astime), we can find a relation between their rates of change. In this case, we say the rates are related,and we can compute one if we know the other.

We proceed as follows:

(a) Identify the independent variable (usually time) on which the other quantities depend and as-sign it a symbol, such as t. Also, assign symbols to the variable quantities that depend on t.

(b) Find an equation that relates the dependent variables.(c) Differentiate both sides of the equation with respect to t (using the chain rule if necessary).(d) Substitute the given information into the related rates equation and solve for the unknown rate.


4. A rocket is launched vertically and is tracked by a radar station located on the ground 5 km from thelaunch pad. Suppose that the elevation angle θ of the line of sight to the rocket is increasing at 3◦ persecond when θ = 60◦. What is the velocity of the rocket at that instant?[link to applet]



5. A man 6 ft tall walks with a speed of 8 ft/s away from a street light that is atop an 18-ft pole. Howfast is the tip of his shadow moving along the ground when he is 100 ft from the light pole?[link to applet]



6. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shorelineand its light makes four revolutions per minute. How fast is the beam of light moving along theshoreline when it is 1 km from P ?[link to applet]



7. Additional Notes

PART 3: DIFFERENTIATION RULES LECTURE 3.10 LINEAR APPROX. AND DIFFERENTIALS 139

3.10 Linear Approximation and Differentials


1. Quote. It is the mark of an instructed mind to rest satisfied with the degree of precision to whichthe nature of the subject admits and not to seek exactness when only an approximation of the truthis possible.(Aristotle, Greek philosopher, 384 BC - 322 BC.)

2. Problem. If f(1) = 4 and f ′(1) = 1 use the linear approximation to f(x) at x = 1 to approximatef(2).

3. Idea. Instead of evaluating f(x) evaluate L(x) where L is the tangent line to the graph of y = f(x)at a known point (a, f(a)) that is close to the point (x, f(x)).

4. Linear Approximation. The linear function

L(x) = f(a) + f ′(a)(x− a)

is called the linearization of f at a.For x close to a we have that

f(x) ≈ L(x) = f(a) + f ′(a)(x− a)

and this approximation is called the linear approximation of f at a.


5. Example. If f(1) = 4 and f ′(1) = 1 use the linear approximation to f(x) at x = 1 to approximatef(2).

6. Example. Use linear approximation to approximate√

37. What is the accuracy of this approxima-tion?


7. Differential. Let f be a function differentiable at x ∈ R. Let ∆x = dx be a (small) given number.The differential dy is defined as

dy = f ′(x)∆x .

8. Important!

f(a+ dx) ≈ L(a+ dx)f(a+ dx) ≈ f(a) + f ′(a)(a+ dx− a)f(a+ dx) ≈ f(a) + f ′(a)dx = f(a) + dy

dy ≈ f(a + dx)− f(a)

Small differential means good approximation.


9. Example. The equatorial radius of the earth is approximately 3960 mi. Suppose that a wire iswrapped tightly around the earth at the equator. Approximately how much must this wire be length-ened if it is to be strung all the way around the earth on poles 10 ft above the ground. (1 mi = 1760yards = 1760 · 3 ft.)

PART 3: DIFFERENTIATION RULES LECTURE 3.11 HYPERBOLIC FUNCTIONS 144

3.11 Hyperbolic Functions


1. Quote. “You could rewrite it in terms of hyperbolic functions, but I don’t know if it’d be easier.”(Posted by Dave at S.O.S. Mathematics CyberBoard, http://www.sosmath.com, )

2. Must Know! Definitions of the hyperbolic functions:

(a) sinh (x) =ex − e−x

2

(b) cosh (x) =ex + e−x

2

(c) tanh (x) =sinhx

coshx

(d) coth (x) =coshx

sinhx

(e) csch (x) =1

sinhx

(f) sech (x) =1

coshx


3. Identities.

(a) sinh(−x) = − sinhx(b) cosh(−x) = coshx(c) cosh2 x− sinh2 x = 1(d) 1− tanh2 x = sech2x(e) sinh(x+ y) = sinhx cosh y + coshx sinh y(f) cosh(x+ y) = coshx cosh y − sinhx sinh y

4. Derivatives of Hyperbolic Functions.

(a) ddx (sinhx) = coshx

(b) ddx (coshx) = sinhx

(c) ddx (tanhx) = sech2x


5. Inverse Hyperbolic Function.

(a)y = sinh−1 x ⇔ sinh y = x

sinh−1 x = ln(x+√x2 + 1), x ∈ R

(b)y = cosh−1 x ⇔ cosh y = x ( where y ≥ 0, x ≥ 1)

cosh−1 x = ln(x+√x2 − 1), x ≥ 1

(c)y = tanh−1 x ⇔ tanh y = x

tanh−1 x =1

2ln

(1 + x

1− x

), −1 < x < 1

6. Derivatives of Inverse Hyperbolic Functions.

(a) ddx (sinh−1 x) =

1√1 + x2

(b) ddx (cosh−1 x) =

1√x2 − 1

(c) ddx (tanh−1 x) =

1

1− x2


7. Example. At what point of the curve y = coshx does the tangent have slope 1?


8. Additional Notes

Part 4

Applications of the Derivative

Image Source: http://commons.wikimedia.org/wiki/File:Mount_Everest_as_seen_from_Drukair.jpg

149

http://commons.wikimedia.org/wiki/File:Mount_Everest_as_seen_from_Drukair.jpg

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.1 MAXIMUM AND MINIMUM VALUES 150

4.1 Maximum and Minimum Values


1. Quote. “I feel the need of attaining the maximum of intensity with the minimum of means. It is thiswhich has led me to give my painting a character of even greater bareness.”(Joan Miró, Catalan-Spanish artist, 1893 - 1983)

2. Definition. A function f has an absolute maximum at c if

f(c) ≥ f(x) for all x ∈ D, the domain of f .

The number f(c) is called the maximum value of f on D.A function f has an absolute minimum at c if

f(c) ≤ f(x) for all x ∈ D, the domain of f .

The number f(c) is called the minimum value of f on D.

3. Definition. A function f has a local maximum at c if

f(c) ≥ f(x) for all x in an open interval, in the domain, containing c .

A function f has a local minimum at c if

f(c) ≤ f(x) for all x in an open interval, in the domain, containing c .

4. Extreme Value Theorem. If f is continuous on a closed interval [a, b], then f attains an absolutemaximum value f(c) and an absolute minimum value f(d) at some numbers c, d ∈ [a, b].


5. Fermat’s Theorem. If f has a local maximum or minimum at c, and f ′(c) exists, then f ′(c) = 0.

6. Examples. Find all local extrema of

(a) f(x) = 3x4 − 16x3 + 18x2, − 1 ≤ x ≤ 4

(b) f(x) = |x|, − 1 < x < 1


7. Definition. A critical number of a function f is a number c in the domain of f such that eitherf ′(c) = 0 or f ′(c) does not exist.

8. Problem. Find the maximum and minimum values of the function

f(x) = x2 + 4x+ 7, −3 ≤ x ≤ 0

9. Closed Interval Method. To find the absolute maximum and minimum values of a continuousfunction f on a closed interval [a, b]:

(a) Find the values of f at the critical numbers of f in (a, b).(b) Find the values of f at the endpoints of the interval.(c) The largest of the values from Step (a) and Step (b) is the absolute maximum value; the smallest

of these values is the absolute minimum value.

10. Examples. Find the maximum and minimum values of the given functions on the indicated closedintervals.

(a) f(x) = x+4

x, x ∈ [1, 4]


(b) g(x) = 2− 3√x, x ∈ [−1, 8]

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.2 THE MEAN VALUE THEOREM 155

4.2 The Mean Value Theorem


1. Quote. “The Mean Value Theorem is the midwife of calculus - not very important or glamorous byitself, but often helping to deliver other theorems that are of major significance.”(Edwin Purcell and Dale Varberg, American mathematicians)

2. Rolle’s Theorem. (Michel Rolle, French mathematician, 1652-1719) Let f be a function that satisfiesthe following three hypotheses:

(a) f is continuous on the closed interval [a, b].(b) f is differentiable on the open interval (a, b).(c) f(a) = f(b).

Then there is a number c in (a, b) such that f ′(c) = 0.

3. Example. Check if the following functions satisfy the hypotheses of Rolle’s theorem.

(a) f(x) = x1/2 − x3/2 on [0, 1].


(b) f(x) = 1− x2/3 on [−1, 1].

4. The Mean Value Theorem. Let f be a function that satisfies the following hypotheses:

(a) f is continuous on the closed interval [a, b].(b) f is differentiable on the open interval (a, b).

Then there is a number c in (a, b) such that

f ′(c) =f(b)− f(a)

b− a

or, equivalently,f(b)− f(a) = f ′(c)(b− a).


5. Example. A car is driving along a rural road where the speed limit is 70 km/h. At 3:00 pm itsodometer reads 18075 km. At 3:18 its reads 18100 km. Prove that the driver violated the speed limitat some instant between 3:00 and 3:18 pm.

6. Show that the equation x4 = x+ 1 has exactly one solution in the interval [1, 2].


7. Must Know! If f ′(x) = 0 for all x in an interval (a, b), then f is constant on (a, b).

8. Fact. If f ′(x) = g′(x) for all x in an interval (a, b), then f − g is constant on (a, b); that is,

f(x) = g(x) + c

where c is a constant.

9. Example. Prove the identity

arcsin

(x− 1x+ 1

)= 2 arctan (

√x)− π

2

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.3 SHAPE OF A GRAPH 160

4.3 How Derivatives Affect the Shape of a Graph


1. Quote. “The spread of civilization may be likened to a fire; First, a feeble spark, next a flickeringflame, then a mighty blaze, ever increasing in speed and power.”(Nikola Tesla, American inventor and engineer, 1856 - 1943)

2. Increasing/Decreasing Test.

(a) If f ′(x) > 0 on an interval, then f is increasing on that interval.(b) If f ′(x) < 0 on an interval, then f is decreasing on that interval.

3. Example. Find the open intervals on the x-axis on which the function

f(x) = 3x4 − 4x3 − 12x2 + 5

is increasing and those on which is decreasing.


4. The First Derivative Test. Suppose that c is a critical number of a continuous function f .

(a) If f ′ changes from positive to negative at c, then f has a local maximum at c.(b) If f ′ changes from negative to positive at c, then f has a local minimum at c.(c) If f ′ does not change sign at c, then f has no local minimum or maximum at c.

5. Example. Find all local extrema of the function

f(x) = 3x4 − 4x3 − 12x2 + 5

6. Examples. Find all local extrema off(x) = |x|


7. Definition. If the graph of f lies above all of its tangent lines on an interval I, then it is calledconcave upward on I. If the graph of f lies below all of its tangents on I, it is called concavedownward on I.

8. Concavity Test.

(a) If f ′′(x) > 0 for all x ∈ I, then the graph of f is concave upward on I.(b) If f ′′(x) < 0 for all x ∈ I, then the graph of f is concave downward on I.

9. Definition. A point P on a curve y = f(x) is called an inflection point if f is continuous there andthe curve changes from concave upward to concave downward or from concave downward to concaveupward at P .


10. The Second Derivative Test. Suppose f ′′ is continuous near c.

(a) If f ′(c) = 0 and f ′′(c) > 0 then f has a local minimum at c.(b) If f ′(c) = 0 and f ′′(c) < 0 then f has a local maximum at c.

11. Example. Sketch the graph of the function

f(x) = 3x4 − 4x3 − 12x2 + 5

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.4 L’HOSPITAL’S RULE 165

4.4 Indeterminate Forms and L’Hospital’s Rule


1. “Proof”. Let a = b.a2 = ab (Multiply both sides by a.)a2 + a2 − 2ab = ab+ a2 − 2ab (Add a2 − 2ab to both sides.)2(a2 − ab) = a2 − ab (Factor the left, and collect like terms on the right.)2 = 1 (Divide both sides by a2 − ab.)

2. Indeterminate Forms.

(a) Indeterminate form of type 00 .

Example. Evaluate limx→0

sin kx

xfor k ∈ R.

(b) Indeterminate form of type ∞∞ .

Example. Evaluate limx→∞

ax+ 1

bx+ 1for a, b ∈ R.


3. L’Hospital’s Rule. Suppose that f and g are differentiable and g′(x) 6= 0 near a (except possibly ata.) Suppose that

limx→a

f(x) = 0 and limx→a

g(x) = 0

or thatlimx→a

f(x) = ±∞ and limx→a

g(x) = ±∞

Thenlimx→a

f(x)

g(x)= limx→a

f ′(x)

g′(x)

if the limit on the right side exists (or is∞ or −∞).

4. Examples. Find

(a) limx→0

ex − 1sin 2x

(b) limx→∞

ex

x2 + x

(c) limx→∞

lnx√x


5. Indeterminate Form 0 · ∞. Find limx→∞

x ln

(x− 1x+ 1

).

6. Indeterminate Form∞−∞. Find limx→0

(1

x− 1

sinx

).

7. Indeterminate Form 00,∞0,1∞.

8. Examples. Find

(a) limx→0

(cosx)1/x2


9. Additional Notes

PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.5 SUMMARY OF CURVE SKETCHI

lecture notes - simon fraser universityvjungic/calculus 1/calculus1.pdf · 2020. 7. 25. · 6.a...

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