math 190 chapter 3 lecture notes 190 chapter 3 notes.pdf · a ladder 10 ft long rests against a...

Math 190

Chapter 3

Lecture Notes

Professor Miguel Ornelas

1

M. Ornelas Math 190 Lecture Notes Section 3.1

Section 3.1

Derivatives of Polynomials and Exponential Functions

Derivative of a Constant Function

d

dx(c) = 0

The Power Rule

In n is any real number, thend

dx(xn) = nxn−1

The Constant Multiple Rule

If c is a constant and f is a differentiable function, then

d

dx[cf(x)] = c

d

dxf(x)

The Sum Rule

If f and g are both differentiable, then

d

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

d

dxf(x) +

d

dxg(x)

The Difference Rule


d

dx[f(x)− g(x)] =

d

dxf(x)− d

dxg(x)

Derivative of the Natural Exponential Function

d

dx(ex) = ex

Differentiate the function.

a. f(x) = 37 b. g(x) =7

4x2 − 3x+ 12

Section 3.1 continued on next page. . . 2

M. Ornelas Math 190 Lecture Notes Section 3.1 (continued)

d. h(t) =4√t− 4et

f. y =

√x+ x

x2

h. k(r) = er + re

e. S(R) = 4πR2

g. F (z) =A+Bz + Cz2

z2

i. y = ex+1 + 1

Find the equation of the tangent line to the curve at the given point.

y = x−√x, (1, 0)



The equation of motion of a particle is

s = t4 − 2t3 + t2 − t, where s is in meters and t is in seconds.

(a) Find the velocity and acceleration as functions of t.

(b) Find the acceleration after 1 s.

Find an equation of the tangent line to the curve y = x4 + 1 that is parallel to the line 32x− y = 15.

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Section 3.2

The Product and Quotient Rules

The Product Rule


d

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

d

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

d

dx[f(x)]

The 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

Differentiation Formulas

d

dx(c) = 0

d

dx(xn) = nxn−1

d

dx(ex) = ex

(cf)′

= cf ′ (f + g)′

= f ′ + g′ (f − g)′

= f ′ − g′

(fg)′

= fg′ + gf ′(f

g

)′=gf ′ − fg′

g2

Differentiate.

a. g(x) = (x+ 2√x)ex b. G(x) =

x2 − 2

2x+ 1



c. y =1

t3 + 2t2 − 1d. h(r) =

aer

b+ er

Find an equation of the tangent line to the given curve at the specified point.

y =1 + x

1 + ex,

(0,

1

2

)

If f(2) = 10 and f ′(x) = x2f(x) for all x, find f ′′(2).



Section 3.3

Derivatives of Trigonometric Functions

Limits

limθ→0

sin θ

θ= 1 lim

θ→0

cos θ − 1

θ= 0

Derivatives of Trigonometric Functions

d

dx(sinx) = cosx

d

dx(cscx) = − cscx cotx

d

dx(cosx) = − sinx

d

dx(secx) = secx tanx

d

dx(tanx) = sec2 x

d

dx(cotx) = − csc2 x

Differentiate.

a. f(x) = x cosx+ 2 tanx

c. y =cosx

1− sinx

b. f(t) =cot t

et

d. f(t) = tet cot t



Find an equation of the tangent line to the curve y = 3x+ 6 cosx at the point (π/3, π + 3).

A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the

wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides

away from the wall, how fast does x change with respect to θ when θ = π/3?

Find the limit.

limθ→0

cos θ − 1

sin θlimx→1

sin(x− 1)

x2 + x− 2



Section 3.4

The Chain Rule

The Chain Rule

If g is differentiable at x and f is differentiable at g(x), then the composite function F = f ◦ g defined by

F (x) = f(g(x)) is differentiable at x and F ′ is given by the product

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

du

dx

The Power Rule Combined with the Chain Rule

If n is any real number and u = g(x) is differentiable, then

d

dx(un) = nun−1

du

dxor

d

dx[g(x)]

n= n [g(x)]

n−1 · g′(x)

Find the derivative of the function.

a. F (x) = (1 + x+ x2)99

c. y =

(x+

1

x

)5

b. f(θ) = sin2 θ

d. U(y) =

(y4 + 1

y2 + 1

)5



d

dx(bx) = bx ln b

e. f(t) = 2t3

f. y = x2e−1/x

Find the 1000th derivative of f(x) = xe−x.



The average blood alcohol concentration BAC of eight male subjects was measured after consumption of 15

mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration

function

C(t) = 0.0225te−0.0467t

where t is measured in minutes after consumption and C is measured in mg/mL.

(a) How rapidly was the BAC increasing after 10 minutes?

(b) How rapidly was it decreasing half an hour later?

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Section 3.5

Implicit Differentiation

Derivatives of Inverse Trigonometric Functions

d

dx(sin−1 x) =

1√1− x2

d

dx

(csc−1 x

)= − 1

x√x2 − 1

d

dx

(cos−1 x

)= − 1√

1− x2d

dx

(sec−1 x

)=

1

x√x2 − 1

d

dx

(tan−1 x

)=

1

1 + x2d

dx

(cot−1 x

)= − 1

1 + x2

Find the derivative of the function.

a. 2x2 + xy − y2 = 2

c. ey sinx = x+ xy

b. xey = x− y

d. x sin y + y sinx = 1



Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

x2 + 2xy + 4y2 = 12, (2, 1) (ellipse)

Find the the derivative of the function. Simplify where possible.

a. y = cos−1(sin−1 t

)b. y = tan−1

(x−

√1 + x2

)

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Section 3.6

Derivatives of Logarithmic Functions

Derivatives of Logarithmic Functions

d

dx(logb x) =

1

x ln b

d

dx(lnx) =

1

x

d

dx(lnu) =

1

u

du

dx

d

dxln |x| = 1

x

Differentiate the function.

a. f(x) = ln(sin2 x)

c. P (v) =ln v

1− v

b. h(x) = ln(x+

√x2 − 1

)

d. y = ln(e−x + xe−x

)



Find an equation of the tangent line to the curve at the given point.

y = x2 lnx, (1, 0)

Use logarithmic differentiation to find the derivative of the function.

a. y =e−x cos2 x

x2 + x+ 1b. y = (sinx)

ln x

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Section 3.7

Rates of Change in the Natural and Social Sciences

If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is s = 80t−16t2.

(a) What is the maximum height reached by the ball?

(b) What is the velocity of the ball when it is 96 ft above the ground on its way up? On its way down?

If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, the volume

V of water remaining in the tank after t minutes is

V = 5000

(1− 1

40t

)2

0 ≤ t ≤ 40

Find the rate at which water is draining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, and (d) 40

min. At what time is the water flowing out the fastest? The slowest?



Newton’s Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body

of mass M is

F =GmM

r2

where G is the gravitational constant and r is the distance between the bodies.

(a) Find dF/dr and explain its meaning. What does the minus sign indicate?

(b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of 2 N/km

when r = 20, 000 km. How fast does this force change when r = 10, 000 km?

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Section 3.9

Related Rates

The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s.

When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?

If a snowball melts so that its surface area decreases at a rate of 1 cm2/min, find the rate at which the

diameter decreases when the diameter is 10 cm.

A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder

slides away from the wall at a rate of 1 ft/s, how fast is the angle between

the ladder and the ground changing when the bottom of the ladder is 6 ft

from the wall?



A baseball diamond is a square with side 90 ft. A batter hits the ball and

runs toward first base with a speed of 24 ft/s.

(a) At what rate is his distance from second base decreasing when he is

halfway to first base?

(b) At what rate is his distance from third base increasing at the same

moment?

A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward

the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he

is 4 m from the building?



Section 3.10

Linear Approximations and Differentials

Linearization of f at a

f(x) ≈ f(a) + f ′(a)(x− a)

L(x) = f(a) + f ′(a)(x− a)

Find the linearization L(x) of the function at a.

f(x) = x3 − x2 + 3, a = −2

Differentials

dy = f ′(x)dx

∆y = f(x+ ∆x)− f(x)

f(a+ dx) ≈ f(a) + dy



Find the differential of the function.

y =1 + 2u

1 + 3u

Use a linear approximation (or differentials) to estimate√

25.07

Use a linear approximation (or differentials) to estimate3√

1001

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math 190 chapter 3 lecture notes 190 chapter 3 notes.pdf · a ladder 10 ft long rests against a...

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