ap calculus - njctlcontent.njctl.org/courses/math/ap-calculus-ab/...recall: the difference quotient...

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AP Calculus Derivatives

20151103

www.njctl.org

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Table of Contents

Rate of Change

Derivative Rules: Power, Constant, Sum/Difference

Continuity vs. Differentiability

Chain Rule

Derivatives of Trig Functions

Implicit Differentiation

Derivatives of Inverse Functions

Derivatives of Logs & e

Slope of a Curve (Instantaneous ROC)

Higher Order Derivatives

Equations of Tangent & Normal Lines

Derivative Rules: Product & QuotientCalculating Derivatives Using Tables

Derivatives of Piecewise & Abs. Value Functions

4

First, let's discuss the importance of Derivatives: Why do we need them?

a) What is the slope of the following function?

b) What is the slope of the line graphed at right?

c) Now, what about the slope of ???

Why are Derivatives Important?

Teacher N

otes

The idea of this slide is to remind students about how much they have learned about slopes in other math classes. Their exposure in previous

math courses has likely been limited to finding slopes of lines only. So when prompted to find the slope of y=x2 this should lead to good discussion among students. Some students may argue that "there is no slope" others may realize "the slope is different at each

point".

5

Exploration into the idea of being locally linear...

Click here to go to the lab titled "Derivatives Exploration: y = x2"

Derivatives Exploration

Teacher N

otes

Lead students through an exploration by having them graph y=x2 (or any curve of their choice) and have them zoom in

slowly by changing their window settings little by little. You want the students to see

that eventually their curve starts to resemble a line. The realization should be that this foreign concept of Derivatives will allow them to be able to find the slopes of curves in particular places, due to the fact

that they are locally linear.URL for Lab: http://njctl.org/courses/math/apcalculusab/derivatives/xsquared

explorationlab/

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Rate of Change

Return to Table of Contents

7

You and your friends take a road trip and leave at 1:00pm, drive 240 miles, and arrive at 5:00pm.How fast were you driving?

Consider the following scenario:

Road Trip!

Teacher N

otes

When students answer 60 mph, ask the following:

• How did you arrive at the answer?• Are you driving 60mph the entire time?

• What does 60mph represent? (they should come up with the words average velocity) *they may say speed, which is

valid at this point.• How would you calculate how fast you

were going at 2:37pm?

8

position

timet1 t2t0 t3

Now, consider the following position vs. time graph:

Position vs. Time

Teacher N

otes

Students can often grasp the concept of derivatives when you relate it to something they

are familiar with, such as velocity.

Discuss with students:• What does the orange line represent? (average velocity over the entire interval)

• What does each green segment represent? (instantaneous velocity at t1 and t2)

9

We will discuss more about average and instantaneous velocity in the next unit, but hopefully it allowed you to

see the difference in calculating slopes at a specific point, rather than over a period of time.

Recap

10

SECANT vs. TANGENT

a

b

x1 x2

y1

y2

A secant line connects 2 points on a curve. The slope of this line is also known as the Average Rate of Change.

A tangent line touches one point on a curve and is known as the Instantaneous Rate of Change.

11

a

b

x1 x2

y1

y2

How would you calculate the slope of the secant line?

Slope of a Secant Line

Answ

er

12

What happens to the slope of the secant line as the point b moves closer to the point a?

a

b

x1 x2

y1

y2

What is the problem with the traditional slope formula when b=a?

Slope of a Secant Line

Teacher N

otes

Allow students to discuss what they think, eventually listening for the conclusion that the secant line resembles the tangent line as

those points get closer together.

Encourage them to observe the fact that the change in x, Δx, gets smaller (approaching 0)

as the point b approaches a.

In reference to the second question, students should note that when b=a, using the

traditional slope formula would result in .

13

Example: Find the average rate of change from x=2 to

x=4 if

It's often useful to find the slope of a secant line, also known as the average rate of change, when using 2 distinct points.

Average Rate of Change

Answ

er

14

1 What is the average rate of change of the function on the interval from to ?

A 3B 2C 1

D 1E 3

Answ

er

15

2 What is the average rate of change of the function on the interval ?

A 14

B 0

C 56

D 56

E 14

Answ

er

16


A B 0 C D E

Answ

er

17


A

B 1

C 2

D

E 0

Answ

er

18

(from the 2007 AP Exam)

5 The wind chill is the temperature, in degrees Fahrenheit, a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity v, in miles per hour. If the air temperature is 32oF, then the wind chill is given by and is valid for 5≤v≤60.

CALCULATOR ALLOWED

Find the average rate of change of W over the interval 5≤v≤60.

Teacher N

otes &

Answer

19

Slope of a Curve(Instantaneous Rate of Change)


20

Recall from the previous unit, we used limits to calculate the instantaneous rate of change using the Difference Quotient.

For example, given , we found an expression to represent the slope at any given point.

Recall: The Difference Quotient

Teacher N

otes

& Answer

This would be a good warm up, and reminder of the difference quotient.

Feel free to work out this problem again, with students. Or simply slide the answer tab out for students to

be reminded of their previous work.

21

The derivative of a function is a formula for the slope of the tangent line to that function at any point x. The process of taking derivatives is called differentiation.We now define the derivative of a function f (x) as

The derivative gives the instantaneous rate of change.In terms of a graph, the derivative gives the slope of the tangent line.

Derivatives

22

Recall the Limits unit, when we discussed alternative representations for the difference quotient as well:

will result in an expression will result in an expression

will result in a number

*where a is constant

Derivatives

23

"f prime of x"

Notation How it's read

"y prime"

"derivative of y with respect to x"

"derivative with respect to x of f(x)"

You may see many different notations for the derivative of a function. Although they look different and are read differently, they still refer to the same concept.

Notation

24

In 1629, mathematician Fermat, was the one to discover that you could calculate the derivative of a function, or the slope a tangent line using the formula:

Formal Definition of a Derivative

25

Example: a) Find the slope at any point, x, of the function b) Use that expression to find the slope of the curve at

Using Fermat's notion of derivatives, we can either find an expression that represents the slope of a curve at any point, x, or if given an xvalue, we can substitute to find the slope at that instant.

Example

Teacher N

otes

& Answer a)

Emphasize to students that the expression they solve for in part a represents the slope of the function

ANYWHERE and doesn't change.

In part b, depending on what value is substituted for x is what results in a different slope at various points.

26

6

A

B

C

D

E

Which expression represents if ?

Answ

er

27

7

A

B

C

D

E

What is the slope of at x=1?

Answ

er

If you notice some students taking much longer than others, it may be that they didn't realize the function was the same from the last slide. With a simple substitution into the correct derivative, they will have

their slope.

28

Find if 8

A

B

C

D

E

Answ

er

29

9

A

B

C

D

E

Find if

Answ

er

If you notice some students taking much longer than others, it may be that they didn't realize the function was the same from the last slide. With a simple substitution into the correct derivative, they will have

their slope.

30

As you may have noticed, derivatives have an important role in mathematics as they allow us to consider what the slope, or rate of change, is of functions other than lines. In the next unit, you will begin to apply the use of derivatives to real world scenarios, understanding how they are even more useful with things such as velocity, acceleration, and

optimization, just to name a few.

Derivatives

31

Derivative Rules:Power, Constant & Sum/Difference


32

Depending on the function, calculating derivatives using Fermat's method with limits can be extremely time consuming. Can you imagine calculating the derivative of using that method?

Or what about ?

Fortunately, there are some "shortcuts" which make taking derivatives much easier!

The AP Exam will still test your knowledge of calculating derivatives using the formal definition (limits), so your energy was not wasted!

Alternate Methods

33

Exploration: Power RuleLet's look back at a few of the derivatives you have calculated already.We found that:

• The derivative of is• The derivative of is• The derivative of is

What observations can you make? Do you notice any shortcuts for

finding these derivatives?

Teacher N

otes

This exploration is meant for students to discover the Power Rule by recognizing patterns. As you show students these

derivatives, allow students to first process quietly, and then encourage discussion in pairs or small groups. Finally, as a class

discuss what conclusions they have made. There are usually some students who have already heard of the Power Rule, or have older siblings/friends that tell them, this is why the quiet reflection time is important for each student to have the opportunity to think.

34

e.g.

e.g.

The Power Rule

*where c is a constant

35

where c is a constant

The Constant Rule

All of these functions have the same derivative. Their derivative is 0.

Why do you think this is?

Think of the meaning of a derivative, and how it applies to the graph of each of these functions.

Teacher N

otes

Lead students in a discussion about what the graphs of each of those functions look like. Hopefully, they will conclude that they are all

equations of horizontal lines. Therefore, no matter where you are

on the graph, the slope of any tangent line will be zero. Hence, the

derivative is zero at any point, regardless of the xvalue.

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e.g.

e.g.

The Sum & Difference Rule

37

Take the derivatives of the following.Practice

Answ

er

38

Sometimes, it takes a little bit of manipulating of the function before applying the Power Rule. Here are 4 scenarios which require an extra step prior to differentiating:

Extra Steps

Teacher N

otes

Distribute then differentiate.

39

10 What is the derivative of ?

A B

C

D E

Answ

er

E

40

11

A

B

C

D

E

Answ

er

C

41

12

A x B 1

C 14

D 0 E 15

What is the derivative of 15?

Answ

er

D

42

13

A

B

C

D

Find if

E

Answ

er

D

43

14

A

B

C

D

Find y' if

Answ

er

C

44

15

A

B

C

D

E

Which expression represents the slope at any point on the curve ? Distribute!HINT

Answ

er

B

45

Find

Calculate

What is the derivative at ?

If asked to find the derivative at a specific point, a question may ask...

Simply find the derivative first, and then substitute the given value for x.

What would happen if you substituted the xvalue first and then tried to take the derivative?

Think...

Derivatives at a Point

Teacher N

otes

46

16 What is the derivative of at ?

A 5B 0C 15 D 5 E 15

Answ

er

C

47

Find 17

A 36 B 144 C 12 D 72 E 24

Answ

er

A

48

18

A 6.5 B 6 C 0 D 3.5

What is the slope of the tangent line at if ?

E 4

Answ

er

E

49

19

A

B

C

D

Find y'(16) if

E Answ

er

B

50



51

You may be wondering.... Can you find the derivative of a derivative!!??

The answer is... YES!

Finding the derivative of a derivative is called the 2nd derivative. Furthermore, taking another derivative would be called the 3rd derivative. So on and so forth.


Teacher N

otes

52

2nd derivative:

3rd derivative:

4th derivative:

nth derivative:

The notation for higher order derivatives is:

Notation

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otes

53

Finding 2nd , 3rd, and higher order derivatives have many practical uses in the real world. In the next unit, you will learn how these derivatives relate to an object's position, velocity, and acceleration. In addition, the 5th derivative is helpful in DNA analysis and population modeling.

Applications of Higher Order Derivatives

54

Find the indicated derivative.

Practice

Teacher N

otes

& Answer

A good discussion question for students is to ask:

How many derivatives must you take depending on the power of x until the derivative reaches 0?

55

20 Find the 3rd derivative of

A B

C

D E

Answ

er

56

21 Find if

A B

C

D E

Answ

er

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22 Find if

A

B

C

D

E

Answ

er

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Find23

A

B

C

D

E

Answ

er

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Find24

A

B

C

D

E

Answ

er

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Find25

A B

C

D E

Answ

er

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Teacher N

otes The reason for placing trig

derivatives prior to product & quotient rule is to allow for more of a variety of problems during these

subsequent sections.

62

For example, if asked to take the derivative of , our previous rules

would not apply.

So far, we have talked about taking derivatives of polynomials, however what about other functions that exist in mathematics?

Next, we will explore derivatives of trigonometric functions!


63


Teacher N

otes

64

Let's take a moment to prove one of these derivatives...

Proof

Teacher N

otes &

Answer

65

Derivatives of Inverse Trig Functions

Teacher N

otes

These derivatives can be very overwhelming for students. Again, encourage flashcards to aid

students with memorization.

While these derivatives will be tested, they are not nearly as critical as the regular trig

derivatives.

The inverse trig derivatives typically show up on the multiple choice portion of the exam.

It is helpful to point out the relationship between positive and negative derivatives to aid them in

memorization.

66

26

A B

C

D E

F

What is the derivative of ?

Answ

er

67

27

A B

C

D E

F


Answ

er

68

28

A B

C

D E

F


Answ

er

69

29

A B

C

D E

F


Answ

er

70

30

A B

C

D E

F


Answ

er

71

31

A B

C

D E

F


Answ

er

72

32

A B

C

D E

F


Answ

er

73

Find33

A 1

B

C

D

E 1

F

Answ

er

It may be necessary to review trig calculations with your students at this time. There are many methods of teaching this,

including the unit circle, special right triangles, or an angle table. Whichever

method students use, they must be efficient and accurate. They cannot rely on a

calculator.

74

Find34

A

B

C

D

E

F

Answ

er

75

Find35

A

B

C

D

E

F

Answ

er

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Find36

A

B

C

D

E

F

Answ

er

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Derivative Rules:Product & Quotient


78

Now... imagine trying to find the derivative of:

Using previous methods of multiplication/distribution, this would be extremely tedious and time consuming!

Need for the Product Rule

79

Fortunately, an alternative method was discovered by the famous calculus mathematician, Gottfried Leibniz, known as the product rule.

Let's take a look at how the product rule works...

The Product Rule

Teacher N

otes

Many teachers find it helpful to have their students say the product rule aloud and repeat it multiple times, as it is

practiced... for example:

"2nd times the derivative of the 1st plus 1st times the derivative of the 2nd."

OR"1st times the derivative of the 2nd plus 2nd

times the derivative of the 1st."

Note: the order in which you add doesn't matter.

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The Product Rule

Notice: You have previously calculated these derivatives by using the distributive property.

The problems above can also be viewed as the product of 2 functions. We can then apply the product rule.

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The Product Rule

using the distributive property using the product rule

Teacher N

otes

It may be helpful for students to practice repeating the product rule. There are various

ways to practice including:"f prime g plus f g prime" or

"2nd times the derivative of the 1st plus 1st times the derivative of the 2nd."

Notice: Both methods yield the same derivative!

82

For example, with the previous function distributing is slightly faster than using the product rule; however, given the function, it may be easier to use the product rule than to try and distribute.

Why use the Product Rule if distribution works just fine?

The complexity of the function will help you determine whether or not to distribute and use the power rule,

versus using the product rule.

Distribution vs. The Product Rule

83

Finding the following derivatives using the product rule.

Practice

Answ

er

84

37

A B

C

D

Answ

er

85

38

A B

C

D

Answ

er

86

39

A

B

C

D

Answ

er

87

Find40

A

B

C

D

Answ

er

88

41

A

B

C

D

Answ

er

89

42

A B

C

D

Answ

er

90

Find43

A B

C

D

Answ

er

Students may not realize they are supposed to be finding the 2nd

derivative. Therefore, using the trig derivative, followed by product

rule.

91

44

True

False

Teacher N

otes FALSE

Students can share/discuss the functions they use to disprove this statement.

92

So far, we have discussed how to take the derivatives of polynomials using the Power Rule, Sum and Difference Rule, and

Constant Rule. We have also discussed how to differentiate trigonometric functions, as well as functions which are comprised

as the product of two functions using the Product Rule.

Next, we will discuss how to approach derivatives of rational functions.

What About Rational Functions?

93

The Quotient Rule

Notice, the problems above can be viewed as the quotient of 2 functions. We can then apply the quotient rule.

Teacher N

otes

Many teachers find it helpful to have their students say the quotient rule aloud and repeat it multiple times, as

it is practiced... for example:

"Bottom times the derivative of the top minus top times the derivative of

the bottom, over the bottom squared."

94

f(x), or "top"

g(x), or "bottom"

FindGiven:

Example

Answ

er

95

Given: Find

Example

Answ

er

This is a good example to explain to students, again, that quotient rule is useful and effective, but in some scenarios it may not be the only way to differentiate (like they saw with distributing

or using product rule).

On this example, they could use the quotient rule, or simplify the rational expression and then find the derivative using power rule.

Perhaps have them try both ways and compare. Clearly, in this particular example, simplifying is quicker than the quotient rule.

96

Now that you have seen the Quotient Rule in action, we can revisit one of the trig derivatives and walk through the proof.

Proof

Answ

er

97

45

A

B

C

D

Differentiate

Answ

er

98

Find46

A

B

C

D

Answ

er

99

Find47

A

B

C

D

Answ

er

Notice: Numerator can be factored, simplifying the equation

to y=x.

100

Find48

A

B

C

D

Answ

er

101

49

A

B

C

D

Differentiate

Answ

er

102

50

A

B

C

D

Find the derivative of

Teacher N

otes &

Answer

This example may confuse students, due to the fact that it contains a fraction. They may attempt to use

quotient rule and get stuck.

The easiest method would be to rewrite the expression and use the power rule to differentiate.

*Note: Quotient rule is still an acceptable method, they could first combine the expression into a

rational expression using a common denominator, or apply quotient rule to the second term only.

103

Calculating Derivatives Using Tables


104

On the AP Exam, in addition to calculating derivatives on your own, you must also be able to use tabular data to find derivatives. These problems are not incredibly difficult, but can be

distracting due to extraneous information.

Derivatives Using Tables

105

Let Calculate

Let's take a look at an example:

The functions f and g are differentiable for all real numbers.

The table above gives values of the functions and their first derivatives at selected values of x.

Example

Answ

er

Using product rule...

106

Let

CalculateThe functions f and g are

differentiable for all real numbers. The table above gives

values of the functions and their first derivatives at selected values of x.

Example

Answ

er

Using power rule & quotient rule...

107

Use the table at right to estimate

Next is another type of question you may encounter on the AP Exam involving tabular data and derivatives.

Derivatives Using Tables

Answ

er

108

Let Calculate

51

A 2

B 10 C 1 D 2

E 30

The functions f and g are differentiable for all real numbers. The table at right gives values of

the functions and their first derivatives at selected values of x.

Answ

er C

109

Let Calculate

52

A 93 B 61 C 75 D 95 E 0

The functions f and g are differentiable for all real numbers. The table at right gives values of

the functions and their first derivatives at selected values of x.

Answ

er A

110

Let Calculate53

A 1.5 B 39 C 32 D 75 E 0

Answ

er B

111

Use the table at right to estimate54

A 2.4 B 3.1 C 32 D 0 E 4

Answ

er E

112

55

A 3.05 B 3.6 C 0 D 0.278 E 0.5

Use the table at right to estimate

Answ

er D

113

Equations of Tangent & Normal Lines


114

Recall from Algebra, that in order to write an equation of a line you either need 2 points, or a slope and a point. If we are asked to find the equation of a tangent line to a curve, our line will touch the curve at a particular point, therefore we will need a slope at that specific point.

Now that we are familiar with calculating derivatives (slopes) we can use our techniques to write these equations of tangent lines.

Writing Equations of Lines

115

First let's consider some basic linear functions...

If asked to write the equation of the tangent line to each of these functions what do you notice?

Equations of Tangent Lines

Teacher N

otes

Have a discussion with students about the fact that a tangent line to

any line is the SAME line!

Let them think about it on their own first, then discuss.

It may help to draw a sketch of a line versus a curve and show how there is only one tangent line to a line, but infinite tangent lines to a curve.

116

Let's try an example: Write an equation for the tangent line to at x=2.

Example

Teacher N

otes &

Answer

117

Write an equation for the tangent line to at .

Example

Answ

er

118

tangent lineat x = 1normal line

at x = 1

y = x2

In addition to finding equations of tangent lines, we also need to find equations of normal lines. Normal lines are defined as the lines which are perpendicular to the tangent line, at the same given point.

How do you suppose we would calculate the slope of a normal line?

Normal Lines

Teacher N

otes

119

Let's try an example: Write an equation for the normal line to at x=2.

Example

Answ

er

120

Example: Write an equation for the normal line to

at .

Example

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er

121

Example: Write an equation for the normal line to

at .

Example

Teacher N

otes &

Answer In order to write an equation we need the point and slope. We can find the

corresponding yvalue for our point by substituting our given xvalue into the equation.

Now, to find the slope, we can calculate the derivative, and substitute our xvalue.

122

Which of the following is the equation of the tangent line to at ?

56

A

B

C

D

E

F

Answ

er

123

Which of the following is the equation of the tangent line toat ?

57

A

B

C

D

E

F

Answ

er

124

Which of the following is the equation of the normal line toat ?

58

A

B

C

D

E

F

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er

125


59

A

B

C

D

E

F

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er

126

Which of the following is the equation of the normal line to at ?

60

A

B

C

D

E

F

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er

127


61

A

B

C

D

E

F

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er

128

Derivatives of Logs & e


129

The next set of functions we will look at are exponential and logarithmic functions, which have their own set

of rules for differentiation.

Exponential and Logarithmic Functions

130

First, let's consider the exponential function

While it appears that Power Rule may be an option, unfortunately it will not apply to this function, because the exponent is not a fixed number, and the base is not the variable.

Exponential Functions

131

By considering a particular value of a, , we are able to see the proof for the derivative of exponential functions.

Note: This proof is based on the fact that e, in the realm of calculus, is the unique number for which

Derivatives of Exponential Functions

Teacher N

otes

132

is the only nontrivial function whose derivative is the same as the function!

cool!

Teacher N

otes

Technically, y=0 is also it's own derivative as well, but does not depend on another variable, so generally people say that is the only one.

Consider asking students if they can think of y=0 before

telling them.


133

At this point, we lack knowledge for the proof of , however, we can prove this derivative when we get to the section on Chain Rule.


134

Derivatives of Logarithmic Functions

Teacher N

otes

Remind students that lnx follows the same rule that logax does, lnx is just

a special case because lne=1.

135

62

A

B

C

D

E

F

Answ

er C

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63

A

B

C

D

E

F

Answ

er C

137

64

A

B

C

D

E

F

Answ

er F

138

65

A

B

C

D

E

F

Find the derivative of

Answ

er A

139

66

A

B

C

D

E

F

Answ

er E

140

Chain Rule


141

Consider the following function:

a) What type of function is this?

b) Would Power Rule or Product Rule be appropriate in finding the derivative of this function?

What About the Following?

Teacher N

otes

a) What type of function is this?

Students may reply, "a composite function" or "a function within a function"

b) Would Power Rule or Product Rule be appropriate in finding the derivative of this function?

Let them discuss this notion and realize that the functions cosx and 2x+3 are not being multiplied, therefore Product Rule won't work. Also, they have only used Power Rule when a single variable is raised to a power, not another function.

142

If

Then

We must apply a new rule when differentiating composite functions known as the Chain Rule.

Chain Rule

Teacher N

otes

143

FindLet's try the Chain Rule on a basic example.

Example

Teacher N

otes &

Answer

Before starting, ask students to identify which function is the "outer" function and which is the

"inner" function.

Hopefully, they will recognize that ( )5 is the outer function and x3+3 is the inner function.

144

Find

Now let's take a look back at the original question and apply Chain Rule... take note of how many "layers" exist in this equation.

Applying Chain Rule

Teacher N

otes &

Answer

145

FindGiven:

Example

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otes &

Answer

146

Find67

A

B

C

D

E

Answ

er B

147

Find68

A

B

C

D

E

Answ

er

148

Find69

A

B

C

D

E

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154

We have already covered derivatives of inverse trig functions, but it is also necessary to calculate the

derivatives of other inverse functions.


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We say that and are invertible if:

Also, if and are invertible then:

Recall the definition of an inverse function..

Inverse Functions

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Taking the derivative of inverse functions requires use of the chain rule, as we can see below.

Fact about inverse functions...

Thus,

Applying chain rule to derive...


Teacher N

otes If you are able to calculate the

inverse function, and then differentiate, that is just fine! The rule is necessary on functions

where the inverse is impossible or very difficult to find.

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Note: As you work through the following problems it is extremely important to pay close attention to notation as you work.

A common error is to forget and/or mix up the inverse sign and derivative sign.

Note the differences:

Be Careful with Notation

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Find the derivative of the inverse of

Example

Answer & Teacher

Notes

In addition, you may wish to show that differentiating the inverse function directly will yield the same answer as using

the rule for inverse derivatives.

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If and find

Example

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Answer

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Suppose and find

Example

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*This time, instead of the value for f' being given to us, we must calculate it, using the derivative function given above.

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Find the derivative of the inverse of 74

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Find the derivative of the inverse of 75

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If and Find 76

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If and Find 77

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165

Continuity vs. Differentiability


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1) f(a) exists

2) exists

3)

Definition of Continuity

In the previous Limits unit, we discussed what must be true for a function to be continuous:

Differentiability requires the same criterion, as well as a few others.

Definition of Continuity

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In order for a function to be considered differentiable, it must contain:

• No discontinuities• No vertical tangent lines• No Corners• No Cusps "sharp points"

Differentiable Functions

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If a function is differentiable, it is also continuous. However, the converse is not true.

Just because a function is continuous does not mean it is differentiable.

What does this mean??? Consider the function:

Notice: If we were asked to find the derivative (slope) at x=0, there is a sharp corner. The slope quickly changes from 1 to 1 as you move closer to x=0. Therefore, this function is not differentiable at x=0.

Differentiability Implies Continuity

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Because there is not one single tangent line that can "balance" at x=0, it is not differentiable at this point.

Another explanation: Imagine zooming in on the function, like we have previously done. The

function must resemble a line ("locally linear") to be differentiable.

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CORNER CUSP

DISCONTINUITY VERTICAL TANGENT

A FUNCTION FAILS TO BE DIFFERENTIABLE IF...

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Types of Discontinuities:

removableinfiniteremovable jump

essential

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+ no sharp points or vertical tangents...

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True

False

If f(x) is continuous on a given interval, it is also differentiable.

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Derivatives of Piecewise & Abs. Value Functions


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Now that we've discussed the criterion for a function to be differentiable, we can look at how to find the derivatives of piecewise and absolute value functions, which often contain sharp

corners, and discontinuities.

Derivatives of Piecewise & Absolute Value Functions

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When calculating derivatives of piecewise functions, the same rules apply for each piece; however, you must also consider the point in which

the function switches from one portion to another.

For a piecewise function to be differentiable EVERYWHERE it must be:• Continuous at all points (equal limits from left and right) • Have equal slopes from left and right

Derivatives of Piecewise & Absolute Value Functions

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Let's first consider the absolute value function...

Visually, we can see that this function is not differentiable at x=0 due to the sharp corner.

Even if it is not differentiable at x=0, we can still find the derivative for the other portions of the graph.

notice: we do not include 0

Derivatives of Absolute Value Functions

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Example: Find the derivative of

It is apparent that every absolute value function will have a sharp point (thus, not being differentiable at that point). But again, we can still find the derivative, discluding the sharp point.

Note: We must first write our function as a piecewise.

Derivatives of Absolute Value Functions

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Find

Let's take a look at some additional piecewise functions.

Derivatives of Piecewise Functions

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Note: 2 is not included in the derivative because although it is continuous, the slopes do not

match from the left and right.

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Is the following piecewise differentiable at ?

Example

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Although the slopes are equivalent at the function is not continuous, there, thus not

differentiable at .

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What values of k and m will make the function differentiable over the interval (0, 5)?

Creating Continuity

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otes & Answer

The method for solving this question is to create a system of equations, because there are 2 unknown variables, k and m.

First, we must ensure the function is continuous at x=3 (equal limits from right and left). To accomplish this, we can plug in 3 to each function and set them equal to each other to create our first equation for the system.

Secondly, we know the slopes must be equal, so taking the derivatives of each function, we can then plug 3 in again and set them equal to each other for our second equation.

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f(x) is continuous at x=2f(x) is differentiable at x=2f(x) is not continuous at x=2f(x) is not differentiable at x=2

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Which of the following is the correct derivative for the function ?

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Choose the correct values for k & m in order for f(x) to be differentiable on the interval (4,9).

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Explicit vs. Implicit Functions

Compare/Contrast the following 2 functions:

vs.

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Allow students to discuss the similarities and differences between the 2 functions.

Often, students realize that one has the y variable isolated and the other does not, which is one basic understanding that helps students understand implicit

functions.

193

So far, all of the derivatives we have taken have been with respect to the variable, x.

e.g.

"derivative of y with respect to x"


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Mathematically, we are actually able to differentiate with respect to any variable, it just requires special attention and notation.

Example: Find y'(t).

"derivative of y with respect to t"


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What happens if our function is in terms of x, but we are asked to find the derivative with respect to a different variable, t?

Example: Find y'(t).

"derivative of y with respect to t"

But, why are these needed?

Due to the fact that we are differentiating with respect to a variable other than what is there, we must include

a dx/dt.


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Students often get confused with the inclusion of a dx/dt or any similar notation. A helpful

explanation is to tell them that technically they have been doing this all along...

196

When a function involves the variables y and x, and y is not isolated on one side of the equation, we must take

additional steps in finding the derivative.

Example: Find

Teacher N

otes &

Answer

Sometimes students tend to stop once they've differentiated across the entire equation. Remind them that the question is asking to find so

they must rearrange their equation until is isolated.


197

Given: Find

1. Differentiate both sides2. Collect all dy/dx to one side3. Factor out dy/dx4. Solve for dy/dx.

Remember! You aren't finished until is isolated.

Example

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1. Differentiate both sides.

2. Collect all dy/dx to one side.

3. Factor out dy/dx.

4. Solve for dy/dx.

198

Find

Find

FindCHALLENGE!

Find

Find

Find

CHALLENGE!

Practice

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Why am I being asked to find the derivative with respect to the variable, t, so often?

Often in Calculus, we are interested in seeing how things change with respect to TIME, hence taking the derivative (which

shows us rate of change) with respect to the variable t.

This will become increasingly more apparent in the next unit when we study Related Rates.

Derivatives with Respect to t

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Now that we have practiced using implicit differentiation, we can

extend the process to find the derivatives at specific points.

Implicit Differentiation at a Point

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Find the slope of the tangent line to the circle given by:at the point

Example

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Plug in values for x and y:

207

For this example, note the benefits of implicit differentiation vs. explicit differentiation. As an optional exercise, you may rework the example for the explicit function:

which is the upper half of the graph.

Then, remember this must be done again if points on the lower half are also desired, given by:

Implicit vs. Explicit Differentiation

208

As a further step in this example, we can now find the equation of the tangent line at the point, (3,4).

Example, Continued

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Example: Find the slope of the graph of

at the point

Example

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Plug in point values:

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94 Find the slope of the tangent line at the point (2, 4) for the equation:

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95 Find the slope of the tangent line at x=3 for the equation:

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Note: Students may need prompting to substitute the xvalue into original function to find the corresponding yvalue to use in derivative.

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96 Find the slope of the tangent line at the point for the equation:

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97 Find the equation of tangent line through point (1, 1) for the equation:

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ap calculus - njctlcontent.njctl.org/courses/math/ap-calculus-ab/...recall: the difference quotient...

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