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AP Calculus Derivatives 2015 11 03 www.njctl.org 2
Table of Contents Rate of Change Slope of a Curve (Instantaneous ROC) Derivative Rules: Power, Constant, Sum/Difference Higher Order Derivatives Derivatives of Trig Functions Derivative Rules: Product & Quotient Calculating Derivatives Using Tables Equations of Tangent & Normal Lines Derivatives of Logs & e Chain Rule Derivatives of Inverse Functions Continuity vs. Differentiability Derivatives of Piecewise & Abs. Value Functions Implicit Differentiation 3
Why are Derivatives Important? 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? Teacher Notes c) Now, what about the slope of??? 4
Derivatives Exploration Exploration into the idea of being locally linear... Click here to go to the lab titled "Derivatives Exploration: y = x 2 " Teacher Notes 5
Rate of Change Return to Table of Contents 6
Road Trip! Consider the following scenario: 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? Teacher Notes 7
Position vs. Time Now, consider the following position vs. time graph: position Teacher Notes t 1 time t 2 t 3 t0 8
Recap 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. 9
SECANT vs. TANGENT A secant line connects 2 b points on a curve. The slope y 2 of this line is also known as the Average Rate of Change. a y 1 A tangent line touches one x 1 x 2 point on a curve and is known as the Instantaneous Rate of Change. 10
Slope of a Secant Line How would you calculate the slope of the secant line? y 2 b y 1 a x 1 x 2 11
Slope of a Secant Line What happens to the slope of the secant line as the point b moves closer to the point a? y 2 y 1 a b Teacher Notes x 1 x 2 What is the problem with the traditional slope formula when b=a? 12
Average Rate of Change 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. Example: Find the average rate of change from x=2 to x=4 if 13
1 What is the average rate of change of the function on the interval from to? A 3 B 2 C 1 D 1 E 3 14
2 What is the average rate of change of the function on the interval? A 14 B 0 C 56 D 56 E 14 15
3 What is the average rate of change of the function on the interval? A B 0 C D E 16
4 What is the average rate of change of the function on the interval? A B 1 C 2 D E 0 17
5 The wind chill is the temperature, in degrees Fahrenheit, a h feels based on the air temperature, in degrees Fahrenheit, a wind velocity v, in miles per hour. If the air temperature of, is then 32 the wind chill is given by and is valid fo 5 v 60. (from the 2007 AP Exam) Find the average rate of change W over of the interval v5 60. CALCULATOR ALLOWED Teacher Notes & 18
Slope of a Curve (Instantaneous Rate of Change) Return to Table of Contents 19
Recall: The Difference Quotient 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. Teacher Notes & 20
Derivatives 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. 21
Derivatives 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 *where a is constant will result in a number 22
Notation 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 How it's read "f prime of x" "y prime" "derivative of y with respect to x" "derivative with respect to x of f(x)" 23
Formal Definition of a Derivative 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: 24
Example 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 x value, we can substitute to find the slope at that instant. Example: a) Find the slope at any point, x, of the function b) Use that expression to find the slope of the curve at Teacher Notes & 25
6 Which expression represents if? A B C D E 26
7 What is the slope of at x=1? A B C D E 27
8 Find if A B C D E 28
9 Find if A B C D E 29
Derivatives 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. 30
Derivative Rules: Power, Constant & Sum/Difference Return to Table of Contents 31
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? Alternate Methods 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! 32
Exploration: Power Rule Let'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 Teacher Notes What observations can you make? Do you notice any shortcuts for finding these derivatives? 33
The Power Rule e.g. e.g. *where c is a constant 34
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 Notes where c is a constant 35
The Sum & Difference Rule e.g. e.g. 36
Practice Take the derivatives of the following. 37
Extra Steps 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: Teacher Notes 38
10 What is the derivative of? A B C D E 39
11 A B C D E 40
12 What is the derivative of 15? A x B 1 C 14 D 0 E 15 41
13 Find if A B C D E 42
14 Find y' if A C B D 43
15 Which expression represents the slope at any point on the curve? Distribute! HINT A B C D E 44
Derivatives at a Point If asked to find the derivative at a specific point, a question may ask... Find Calculate What is the derivative at? Simply find the derivative first, and then substitute the given value for x. Think... What would happen if you substituted the x value first and then tried to take the derivative? Teacher Notes 45
16 What is the derivative of at? A 5 B 0 C 15 D 5 E 15 46
17 Find A 36 B 144 C 12 D 72 E 24 47
18 What is the slope of the tangent line at if? A 6.5 B 6 C 0 D 3.5 E 4 48
19 Find y'(16) if A C E B D 49
Higher Order Derivatives Return to Table of Contents 50
Higher Order Derivatives 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 2 nd derivative. Furthermore, taking another derivative would be called the 3 rd derivative. So on and so forth. Teacher Notes 51
Notation The notation for higher order derivatives is: 2 nd derivative: 3 rd derivative: Teacher Notes 4th derivative: n th derivative: 52
Applications of Higher Order Derivatives Finding 2 nd, 3 rd, 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 5 th derivative is helpful in DNA analysis and population modeling. 53
Find the indicated derivative. Practice Teacher Notes & 54
20 Find the 3rd derivative of A B C D E 55
21 Find if A B C D E 56
22 Find if A B C D E 57
23 Find A B C D E 58
24 Find A B C D E 59
25 Find A B C D E 60
Derivatives of Trig Functions Return to Table of Contents Teacher Notes 61
Derivatives of Trig Functions 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! For example, if asked to take the derivative of, our previous rules would not apply. 62
Derivatives of Trig Functions Teacher Notes 63
Proof Let's take a moment to prove one of these derivatives... Teacher Notes & 64
Derivatives of Inverse Trig Functions Teacher Notes 65
26 What is the derivative of? A B C D E F 66
27 What is the derivative of? A B C D E F 67
28 What is the derivative of? A B C D E F 68
29 What is the derivative of? A B C D E F 69
30 What is the derivative of? A B C D E F 70
31 What is the derivative of? A B C D E F 71
32 What is the derivative of? A B C D E F 72
33 Find A 1 B D E 1 C F 73
34 Find A B D E C F 74
35 Find A B D E C F 75
36 Find A B D E C F 76
Derivative Rules: Product & Quotient Return to Table of Contents 77
Need for the Product Rule Now... imagine trying to find the derivative of: Using previous methods of multiplication/distribution, this would be extremely tedious and time consuming! 78
The Product Rule 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... Teacher Notes 79
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. 80
The Product Rule using the distributive property using the product rule Teacher Notes 81
Distribution vs. The Product Rule 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. 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. 82
Practice Finding the following derivatives using the product rule. 83
37 A B C D 84
38 A B C D 85
39 A B C D 86
40 Find A B C D 87
41 A B C D 88
42 A B C D 89
43 Find A B C D 90
44 True False Teacher Notes 91
What About Rational Functions? 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. 92
The Quotient Rule Notice, the problems above can be viewed as the quotient of 2 functions. We can then apply the quotient rule. Teacher Notes 93
Example Given: Find f(x), or "top" g(x), or "bottom" 94
Example Given: Find 95
Proof Now that you have seen the Quotient Rule in action, we can revisit one of the trig derivatives and walk through the proof. 96
45 Differentiate A B C D 97
46 Find A B C D 98
47 Find A B C D 99
48 Find A B C D 100
49 Differentiate A B C D 101
50 Find the derivative of A B C Teacher Notes & D 102
Calculating Derivatives Using Tables Return to Table of Contents 103
Derivatives Using Tables 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. 104
Let's take a look at an example: Example Let Calculate 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. 105
Example Let Calculate 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. 106
Derivatives Using Tables Next is another type of question you may encounter on the AP Exam involving tabular data and derivatives. Use the table at right to estimate 107
51 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. Let Calculate A 2 B 10 C 1 D 2 E 30 108
52 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. Let Calculate A 93 B 61 C 75 D 95 E 0 109
53 Let Calculate A 1.5 B 39 C 32 D 75 E 0 110
54 Use the table at right to estimate A 2.4 B 3.1 C 32 D 0 E 4 111
55 Use the table at right to estimate A 3.05 B 3.6 C 0 D 0.278 E 0.5 112
Equations of Tangent & Normal Lines Return to Table of Contents 113
Writing Equations of Lines 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. 114
Equations of Tangent Lines First let's consider some basic linear functions... Teacher Notes If asked to write the equation of the tangent line to each of these functions what do you notice? 115
Example Let's try an example: Write an equation for the tangent line to at x=2. Teacher Notes & 116
Example Write an equation for the tangent line to at. 117
y = x 2 Normal Lines 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. normal line at x = 1 tangent line at x = 1 Teacher Notes How do you suppose we would calculate the slope of a normal line? 118
Example Let's try an example: Write an equation for the normal line to at x=2. 119
Example Example: Write an equation for the normal line to at. 120
Example: Write an equation for the normal line to at. Example Teacher Notes & 121
56 Which of the following is the equation of the tangent line to at? A B C D E F 122
57 Which of the following is the equation of the tangent line to at? A B C D E F 123
58 Which of the following is the equation of the normal line to at? A B C D E F 124
59 Which of the following is the equation of the tangent line to at? A B C D E F 125
60 Which of the following is the equation of the normal line to at? A B C D E F 126
61 Which of the following is the equation of the tangent line to at? A B C D E F 127
Derivatives of Logs & e Return to Table of Contents 128
Exponential and Logarithmic Functions The next set of functions we will look at are exponential and logarithmic functions, which have their own set of rules for differentiation. 129
Exponential Functions 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. 130
Derivatives of Exponential Functions 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 Teacher Notes 131
Derivatives of Exponential Functions cool! is the only nontrivial function whose derivative is the same as the function! Teacher Notes 132
Derivatives of Exponential Functions 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. 133
Derivatives of Logarithmic Functions Teacher Notes 134
62 A B D E C F 135
63 A B D E C F 136
64 A B D E C F 137
65 Find the derivative of A B D E C F 138
66 A B D E C F 139
Chain Rule Return to Table of Contents 140
What About the Following? Consider the following function: a) What type of function is this? Teacher Notes b) Would Power Rule or Product Rule be appropriate in finding the derivative of this function? 141
Chain Rule We must apply a new rule when differentiating composite functions known as the Chain Rule. If Teacher Notes Then 142
Example Let's try the Chain Rule on a basic example. Find Teacher Notes & 143
Applying Chain Rule 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. Find Teacher Notes & 144
Example Given: Find Teacher Notes & 145
67 Find A B C D E 146
68 Find A B C D E 147
69 Find A B C D E 148
70 Find A B C D E 149
71 Find A B C D E 150
72 Find A B C D 151
73 Find A B C D E 152
Derivatives of Inverse Functions Return to Table of Contents 153
Derivatives of Inverse Functions We have already covered derivatives of inverse trig functions, but it is also necessary to calculate the derivatives of other inverse functions. 154
Inverse Functions Recall the definition of an inverse function.. We say that and are invertible if: Teacher Notes Also, if and are invertible then: 155
Derivatives of Inverse Functions Taking the derivative of inverse functions requires use of the chain rule, as we can see below. Fact about inverse functions... Teacher Notes Applying chain rule to derive... Thus, 156
Be Careful with Notation 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: 157
Find the derivative of the inverse of Example & Teacher Notes 158
Example If and find Teacher Notes & 159
Example Suppose and find 160
74 Find the derivative of the inverse of A B C D 161
75 Find the derivative of the inverse of A B C D 162
76 If and Find A B C D 163
77 If and Find A B C D 164
Continuity vs. Differentiability Return to Table of Contents 165
Definition of Continuity In the previous Limits unit, we discussed what must be true for a function to be continuous: Definition of Continuity 1) f(a) exists 2) exists 3) Differentiability requires the same criterion, as well as a few others. 166
Differentiable Functions In order for a function to be considered differentiable, it must contain: No discontinuities No vertical tangent lines No Corners No Cusps "sharp points" 167
Differentiability Implies Continuity 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: Teacher Notes 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. 168
A FUNCTION FAILS TO BE DIFFERENTIABLE IF... CORNER CUSP DISCONTINUITY VERTICAL TANGENT 169
Types of Discontinuities: removable removable jump infinite essential 170
+ no sharp points or vertical tangents... 171
78 Choose all values of x where f(x) is not differentiable. A B C D E F G 172
79 Choose all values of x where f(x) is not differentiable. A B C D E F G 173
80 Choose all values of x where f(x) is not differentiable. A B C D E F G 174
81 If f(x) is continuous on a given interval, it is also differentiable. True False 175
Derivatives of Piecewise & Abs. Value Functions Return to Table of Contents 176
Derivatives of Piecewise & Absolute Value Functions 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. 177
Derivatives of Piecewise & Absolute Value Functions 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 178
Derivatives of Absolute Value Functions 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. Teacher Notes notice: we do not include 0 179
Derivatives of Absolute Value Functions 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. Example: Find the derivative of Note: We must first write our function as a piecewise. Teacher Notes 180
Derivatives of Piecewise Functions Let's take a look at some additional piecewise functions. Find 181
Example Is the following piecewise differentiable at? 182
Creating Continuity What values of k and m will make the function differentiable over the interval (0, 5)? Teacher Notes & 183
82 A B C D f(x) is continuous at x= 2 f(x) is differentiable at x= 2 f(x) is not continuous at x= 2 f(x) is not differentiable at x= 2 184
83 A B C D f(x) is continuous at x=0 f(x) is differentiable at x=0 f(x) is not continuous at x=0 f(x) is not differentiable at x=0 185
84 A B C D f(x) is continuous at x=1 f(x) is differentiable at x=1 f(x) is not continuous at x=1 f(x) is not differentiable at x=1 186
85 A B C D f(x) is continuous at x=0 f(x) is differentiable at x=0 f(x) is not continuous at x=0 f(x) is not differentiable at x=0 187
86 A B C D f(x) is continuous at x= 5 f(x) is differentiable at x= 5 f(x) is not continuous at x= 5 f(x) is not differentiable at x= 5 188
87 Which of the following is the correct derivative for the function? A D B E C F 189
88 Choose the correct values for k & m in order for f(x) to be differentiable on the interval ( 4,9). A B C D E F G H 190
Implicit Differentiation Return to Table of Contents 191
Explicit vs. Implicit Functions Compare/Contrast the following 2 functions: vs. Teacher Notes 192
Implicit Differentiation 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" 193
Implicit Differentiation 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" 194
Implicit Differentiation 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: "derivative of y with respect to t" Find y'(t). Teacher Notes 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. 195
Implicit Differentiation 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 Notes & 196
Example Remember! You aren't finished until is isolated. 1. Differentiate both sides 2. Collect all dy/dx to one side 3. Factor out dy/dx 4. Solve for dy/dx. Given: Find 197
Practice Find Find Find Find CHALLENGE! Find CHALLENGE! Find 198
Derivatives with Respect to t 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. 199
89 Find A B C D 200
90 Find A B C D 201
91 Find A B C D 202
92 Find A B C D 203
93 Find A B C D 204
Implicit Differentiation at a Point Now that we have practiced using implicit differentiation, we can extend the process to find the derivatives at specific points. 205
Example Find the slope of the tangent line to the circle given by: at the point 206
Implicit vs. Explicit Differentiation 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: 207
Example, Continued As a further step in this example, we can now find the equation of the tangent line at the point, (3,4). 208
Example Example: Find the slope of the graph of at the point 209
94 Find the slope of the tangent line at the point (2, 4) for the equation: A B C D 210
95 Find the slope of the tangent line at x=3 for the equation: A B C D 211
96 Find the slope of the tangent line at the point for the equation: A B C D 212
97 Find the equation of tangent line through point (1, 1) for the equation: A B C D 213