AP Calculus. Slide 1 / 213 Slide 2 / 213. Slide 3 / 213. Slide 4 / 213. Slide 4 (Answer) / 213 Slide 5 / 213. Derivatives. Derivatives Exploration

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1 Slide 1 / 213 Slide 2 / 213 AP Calculus Derivatives Slide 3 / 213 Table of Contents Slide 4 / 213 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 Slide 4 (Answer) / 213 Slide 5 / 213 Derivatives Exploration Exploration into the idea of being locally linear... Click here to go to the lab titled "Derivatives Exploration: y = x 2 "

2 Slide 5 (Answer) / 213 Derivatives Exploration Lead students through an exploration by having them graph y=x 2 (or any curve of their choice) and have them zoom in slowly by changing their window settings Exploration into the idea little of being by little. locally You want linear... the students to see Teacher Notes that eventually their curve starts to resemble a line. The realization should be that this foreign concept of Derivatives will Click here to go allow to them lab to titled be able "Derivatives to find the slopes of Exploration: y = curves x 2 " in particular places, due to the fact that they are locally linear. URL for Lab: ap-calculus-ab/derivatives/x-squaredexploration-lab/ is a teacher notes pull [This object tab] Slide 6 / 213 Rate of Change Return to Table of Contents Consider the following scenario: Slide 7 / 213 Road Trip! Consider the following scenario: Slide 7 (Answer) / 213 Road Trip! 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 When You students and your answer friends 60 mph, take a ask road the trip and leave at 1:00pm, following: drive 240 miles, and arrive at 5:00pm. How fast were you driving? 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? [This object is a teacher notes pull tab] Slide 8 / 213 Slide 8 (Answer) / 213 Now, consider the following position vs. time graph: position Position vs. Time Teacher Notes Now, Students consider can the often following grasp position the concept vs. time of graph: derivatives when you relate it to something they are familiar with, such as velocity. position Position vs. Time Discuss with students: What does the orange line represent? (average velocity over the entire interval) What does each green segment represent? [This object is a teacher notes pull tab] (instantaneous velocity at t 1 and t 2) t0 t 1 time t 2 t 3 t0 t 1 time t 2 t 3

3 Slide 9 / 213 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. Slide 10 / 213 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. Slide 11 / 213 Slide 11 (Answer) / 213 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 Slide 12 / 213 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 x 1 x 2 Slide 12 (Answer) / 213 Slope of a Secant Line Allow students to discuss what they think, What eventually happens listening to the slope for the of the conclusion secant that line the as the point b moves secant closer line to the resembles point a? the tangent line as those points get closer together. Teacher Notes b Encourage y 2 them to observe the fact that the change in x, #x, gets smaller (approaching 0) as the point b approaches a a. y 1 In reference to the second question, students should note that when x b=a, 1 using x 2 the traditional slope formula would result in. [This object is a teacher notes pull tab] What is the problem with the traditional slope formula when b=a? What is the problem with the traditional slope formula when b=a?

4 Slide 13 / 213 Slide 13 (Answer) / 213 Slide 14 / 213 Slide 14 (Answer) / 213 Slide 15 / 213 Slide 15 (Answer) / 213

5 Slide 16 / 213 Slide 16 (Answer) / 213 Slide 17 / 213 Slide 17 (Answer) / 213 Slide 18 / 213 Slide 18 (Answer) / 213

6 Slide 19 / 213 Slide 20 / 213 Slope of a Curve (Instantaneous Rate of Change) Return to Table of Contents Slide 20 (Answer) / 213 Slide 21 / 213 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. Slide 22 / 213 Slide 23 / 213 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)"

7 Slide 24 / 213 Slide 25 / 213 Slide 25 (Answer) / 213 Slide 26 / 213 Slide 26 (Answer) / 213 Slide 27 / 213

8 Slide 27 (Answer) / 213 Slide 28 / 213 Slide 28 (Answer) / 213 Slide 29 / 213 Slide 29 (Answer) / 213 Slide 30 / 213 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.

9 Slide 31 / 213 Slide 32 / 213 Derivative Rules: Power, Constant & Sum/Difference Return to Table of Contents Slide 33 / 213 Slide 33 (Answer) / 213 Slide 34 / 213 Slide 35 / 213 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. where c is a constant

10 Slide 35 (Answer) / 213 Slide 36 / 213 The Constant Rule Teacher Notes where c is a constant All of these functions have the same Lead students in a discussion derivative. Their derivative is 0. about what the graphs of each of those functions look like. Hopefully, they will Why conclude do you that think they this are is? all equations of horizontal lines. Therefore, Think of no the matter meaning where of you a derivative, are on and the how graph, it applies the slope to the of any graph of [This object is a teacher notes pull tab] tangent each line of these will be functions. zero. Hence, the derivative is zero at any point, regardless of the x-value. Slide 37 / 213 Slide 37 (Answer) / 213 Slide 38 / 213 Slide 38 (Answer) / 213

11 Slide 39 / 213 Slide 39 (Answer) / 213 Slide 40 / 213 Slide 40 (Answer) / A B C A B C Answer C D D [This object is a pull tab] E E Slide 41 / 213 Slide 41 (Answer) / What is the derivative of 15? 12 What is the derivative of 15? A x B 1 A x B 1 Answer D C 14 C 14 [This object is a pull tab] D 0 D 0 E -15 E -15

12 Slide 42 / 213 Slide 42 (Answer) / 213 Slide 43 / 213 Slide 43 (Answer) / Find y' if 14 Find y' if A C A C Answer C B D B D [This object is a pull tab] Slide 44 / 213 Slide 44 (Answer) / 213

13 Slide 45 / 213 Slide 45 (Answer) / 213 Slide 46 / 213 Slide 46 (Answer) / 213 Slide 47 / 213 Slide 47 (Answer) / 213

14 Slide 48 / 213 Slide 48 (Answer) / 213 Slide 49 / 213 Slide 49 (Answer) / Find y'(16) if 19 Find y'(16) if A C E A C Answer E B B D B D [This object is a pull tab] Slide 50 / 213 Slide 51 / 213 Higher Order Derivatives You may be wondering... Can you find the derivative of a derivative!!?? Higher Order Derivatives Return to Table of Contents 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.

15 Slide 51 (Answer) / 213 Slide 52 / 213 Higher Order Derivatives Teacher Notes You may be wondering... Can you find the You derivative may want of a to derivative!!?? mention to students that each derivative they take must be continuous in order The answer is... YES! to keep taking the next derivative. Continuity and differentiability will Finding the derivative of be a discussed derivative later, is called but it the is helpful 2 nd derivative. Furthermore, taking another derivative to mention would here. be called the 3 rd [This object is a teacher notes pull tab] derivative. So on and so forth. Slide 52 (Answer) / 213 Slide 53 / 213 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. Slide 54 / 213 Slide 54 (Answer) / 213

16 Slide 55 / 213 Slide 55 (Answer) / 213 Slide 56 / 213 Slide 56 (Answer) / 213 Slide 57 / 213 Slide 57 (Answer) / Find if 22 Find if A A B B Answer A C C D D [This object is a pull tab] E E

17 Slide 58 / 213 Slide 58 (Answer) / 213 Slide 59 / 213 Slide 59 (Answer) / 213 Slide 60 / 213 Slide 60 (Answer) / 213

18 Slide 61 / 213 Slide 61 (Answer) / 213 Derivatives of Trig Functions Return to Table of Contents Teacher Notes Derivatives of Trig Functions 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. Return to [This object is a teacher notes pull tab] Table of Contents Slide 62 / 213 Slide 63 / 213 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. Slide 63 (Answer) / 213 Slide 64 / 213 Proof Let's take a moment to prove one of these derivatives...

19 Slide 64 (Answer) / 213 Proof g used. Let's take a moment to prove one of these derivatives... derivatives,. There are es. Slide 65 / 213 m identity Slide 65 (Answer) / 213 Slide 66 / 213 Slide 66 (Answer) / 213 Slide 67 / 213

20 Slide 67 (Answer) / 213 Slide 68 / 213 Slide 68 (Answer) / 213 Slide 69 / 213 Slide 69 (Answer) / 213 Slide 70 / 213

21 Slide 70 (Answer) / 213 Slide 71 / 213 Slide 71 (Answer) / 213 Slide 72 / 213 Slide 72 (Answer) / 213 Slide 73 / 213

22 Slide 73 (Answer) / 213 Slide 74 / Find A B D E C F Slide 74 (Answer) / 213 Slide 75 / Find A B D E Answer D C F [This object is a pull tab] Slide 75 (Answer) / 213 Slide 76 / Find A B D E C F

23 Slide 76 (Answer) / 213 Slide 77 / Find A B D E Answer E Derivative Rules: Product & Quotient C F [This object is a pull tab] Return to Table of Contents Slide 78 / 213 Slide 79 / 213 Slide 79 (Answer) / 213 Slide 80 / 213

24 Slide 81 / 213 Slide 81 (Answer) / 213 Slide 82 / 213 Slide 83 / 213 Slide 83 (Answer) / 213 Slide 84 / 213

25 Slide 84 (Answer) / 213 Slide 85 / 213 Slide 85 (Answer) / 213 Slide 86 / 213 Slide 86 (Answer) / 213 Slide 87 / 213

26 Slide 87 (Answer) / 213 Slide 88 / 213 Slide 88 (Answer) / 213 Slide 89 / 213 Slide 89 (Answer) / 213 Slide 90 / 213

27 Slide 90 (Answer) / 213 Slide 91 / True False Slide 91 (Answer) / 213 Slide 92 / True False Teacher Notes FALSE Students can share/discuss the functions they use to disprove this statement. 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. [This object is a teacher notes pull tab] Next, we will discuss how to approach derivatives of rational functions. Slide 93 / 213 Slide 93 (Answer) / 213

28 Slide 94 / 213 Slide 94 (Answer) / 213 Example Example Given: Find Given: Find f(x), or "top" g(x), or "bottom" Answer f(x), or "top" g(x), or "bottom" [This object is a pull tab] Slide 95 / 213 Slide 95 (Answer) / 213 Example Given: Find Slide 96 / 213 Slide 96 (Answer) / 213 Proof Now that you have seen the Quotient Rule in action, we can revisit one of the trig derivatives and walk through the proof.

29 Slide 97 / 213 Slide 97 (Answer) / 213 Slide 98 / 213 Slide 98 (Answer) / 213 Slide 99 / 213 Slide 99 (Answer) / 213

30 Slide 100 / 213 Slide 100 (Answer) / 213 Slide 101 / 213 Slide 101 (Answer) / 213 Slide 102 / 213 Slide 102 (Answer) / 213

31 Slide 103 / 213 Slide 104 / 213 Derivatives Using Tables Calculating 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. Return to Table of Contents Slide 105 / 213 Slide 106 / 213 Slide 107 / 213 Slide 108 / 213

32 Slide 109 / 213 Slide 110 / 213 Slide 110 (Answer) / 213 Slide 111 / 213 Slide 112 / 213 Slide 113 / 213 Equations of Tangent & Normal Lines Return to Table of Contents

33 Slide 114 / 213 Slide 115 / 213 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. Slide 115 (Answer) / 213 Slide 116 / 213 Slide 116 (Answer) / 213 Slide 117 / 213

34 Slide 117 (Answer) / 213 Slide 118 / 213 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. y = x 2 normal line at x = 1 tangent line at x = 1 How do you suppose we would calculate the slope of a normal line? Slide 118 (Answer) / 213 Slide 119 / 213 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. This y question = x 2 is meant to be in general, for any normal line, not necessarily this specific curve. tangent line normal line at x = 1 at Allow x = 1 students to discuss how to come up with the slope of the normal line, hoping they make the connection that perpendicular lines [This have object is a opposite teacher notes reciprocal pull tab] slopes. Teacher Notes How do you suppose we would calculate the slope of a normal line? Slide 119 (Answer) / 213 Slide 120 / 213

35 Slide 120 (Answer) / 213 Slide 121 / 213 Slide 121 (Answer) / 213 Slide 122 / 213 Slide 122 (Answer) / 213 Slide 123 / 213

36 Slide 123 (Answer) / 213 Slide 124 / 213 Slide 124 (Answer) / 213 Slide 125 / 213 Slide 125 (Answer) / 213 Slide 126 / 213

37 Slide 126 (Answer) / 213 Slide 127 / 213 Slide 127 (Answer) / 213 Slide 128 / 213 Derivatives of Logs & e Return to Table of Contents Slide 129 / 213 Slide 130 / 213 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.

38 Slide 131 / 213 Slide 131 (Answer) / 213 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 Slide 132 / 213 Derivatives of Exponential Functions Slide 132 (Answer) / 213 Derivatives of Exponential Functions cool! is the only nontrivial function whose derivative is the same as the function! Teacher Notes 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 is the think only of y=0 nontrivial before function whose derivative telling them. is the same as the function! cool! [This object is a teacher notes pull tab] Slide 133 / 213 Slide 134 / 213

39 Slide 134 (Answer) / 213 Slide 135 / A B D E C F Slide 135 (Answer) / 213 Slide 136 / A B D E Answer C C F [This object is a pull tab] Slide 136 (Answer) / 213 Slide 137 / 213

40 Slide 137 (Answer) / 213 Slide 138 / 213 Slide 138 (Answer) / 213 Slide 139 / 213 Slide 139 (Answer) / 213 Slide 140 / 213 Chain Rule Return to Table of Contents

41 Slide 141 / 213 Slide 141 (Answer) / 213 Slide 142 / 213 Slide 142 (Answer) / 213 Slide 143 / 213 Slide 143 (Answer) / 213

42 Slide 144 / 213 Slide 144 (Answer) / 213 Slide 145 / 213 Slide 145 (Answer) / 213 Slide 146 / 213 Slide 146 (Answer) / 213

43 Slide 147 / 213 Slide 147 (Answer) / 213 Slide 148 / 213 Slide 148 (Answer) / 213 Slide 149 / 213 Slide 149 (Answer) / 213

44 Slide 150 / 213 Slide 150 (Answer) / 213 Slide 151 / 213 Slide 151 (Answer) / 213 Slide 152 / 213 Slide 152 (Answer) / 213

45 Slide 153 / 213 Slide 154 / 213 Derivatives of Inverse Functions 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. Return to Table of Contents Slide 155 / 213 Slide 155 (Answer) / 213 Slide 156 / 213 Slide 156 (Answer) / 213

46 Slide 157 / 213 Slide 158 / 213 Slide 158 (Answer) / 213 Slide 159 / 213 Example If and find Slide 159 (Answer) / 213 Slide 160 / 213

47 Slide 160 (Answer) / 213 Slide 161 / 213 Slide 161 (Answer) / 213 Slide 162 / 213 Slide 162 (Answer) / 213 Slide 163 / 213

48 Slide 163 (Answer) / 213 Slide 164 / 213 Slide 164 (Answer) / 213 Slide 165 / 213 Continuity vs. Differentiability Return to Table of Contents Slide 166 / 213 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 Slide 167 / 213 Differentiable Functions In order for a function to be considered differentiable, it must contain: No discontinuities No vertical tangent lines No Corners "sharp points" No Cusps 3) Differentiability requires the same criterion, as well as a few others.

49 Slide 168 / 213 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: 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. Teacher Notes Slide 168 (Answer) / 213 Differentiability Implies Continuity Another way to explain to this to students is to draw several tangent lines at x=0 and show that they all have different slopes. If a function is differentiable, it is also continuous. However, the converse is not true. Just because Because a function there is is not continuous one single does tangent not mean line that it is can differentiable. "balance" at x=0, it is not differentiable at this point. What does this mean??? Consider the function: Another explanation: Imagine zooming in on the function, like we have previously done. The function must resemble a line [This object is a teacher notes pull tab] Notice: ("locally If we linear") were asked to to find the be differentiable. 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. CORNER Slide 169 / 213 A FUNCTION FAILS TO BE DIFFERENTIABLE IF... CUSP Slide 170 / 213 Types of Discontinuities: removable removable jump infinite DISCONTINUITY VERTICAL TANGENT essential Slide 171 / 213 Slide 172 / 213

50 Slide 172 (Answer) / 213 Slide 173 / 213 Slide 173 (Answer) / 213 Slide 174 / 213 Slide 174 (Answer) / 213 Slide 175 / If f(x) is continuous on a given interval, it is also differentiab True False

51 Slide 175 (Answer) / 213 Slide 176 / If f(x) is continuous on a given interval, it is also differentiab True False Answer False Derivatives of Piecewise & Abs. Value Functions [This object is a pull tab] Return to Table of Contents Slide 177 / 213 Slide 178 / 213 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. 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 Slide 179 / 213 Slide 179 (Answer) / 213

52 Slide 180 / 213 Slide 180 (Answer) / 213 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. Slide 181 / 213 Slide 181 (Answer) / 213 Slide 182 / 213 Slide 182 (Answer) / 213

53 Slide 183 / 213 Slide 183 (Answer) / 213 Slide 184 / 213 Slide 184 (Answer) / 213 Slide 185 / 213 Slide 185 (Answer) / 213

54 Slide 186 / 213 Slide 186 (Answer) / 213 Slide 187 / 213 Slide 187 (Answer) / 213 Slide 188 / 213 Slide 188 (Answer) / 213

55 Slide 189 / 213 Slide 189 (Answer) / 213 Slide 190 / 213 Slide 190 (Answer) / 213 Slide 191 / 213 Slide 192 / 213 Implicit Differentiation Return to Table of Contents

56 Slide 192 (Answer) / 213 Slide 193 / 213 Slide 194 / 213 Slide 195 / 213 Slide 195 (Answer) / 213 Slide 196 / 213

57 Slide 196 (Answer) / 213 Slide 197 / 213 Slide 197 (Answer) / 213 Slide 198 / 213 Practice Find Find Find Find CHALLENGE! Find CHALLENGE! Find Slide 198 (Answer) / 213 Slide 199 / 213 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.

58 Slide 200 / 213 Slide 200 (Answer) / 213 Slide 201 / 213 Slide 201 (Answer) / 213 Slide 202 / 213 Slide 202 (Answer) / 213

59 Slide 203 / 213 Slide 203 (Answer) / 213 Slide 204 / 213 Slide 204 (Answer) / 213 Slide 205 / 213 Implicit Differentiation at a Point Now that we have practiced using implicit differentiation, we can Slide 206 / 213 Example Find the slope of the tangent line to the circle given by: at the point extend the process to find the derivatives at specific points.

60 Slide 206 (Answer) / 213 Example 3. Factor out dy/dx Find the slope of the tangent line to the circle 4. Solve given for dy/dx. by: Answer Find at the point Plug in values for x and y: Recall: 1. Differentiate both sides 2. Collect all dy/dx to one side Slide 207 / 213 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. [This object is a pull tab] Then, remember this must be done again if points on the lower half are also desired, given by: Slide 208 / 213 Example, Continued As a further step in this example, we can now find the equation of the tangent line at the point, (3,4). Slide 208 (Answer) / 213 Example, Continued Find the equation of the tangent line at (3,4): As a further step in this example, we can now find the equation From before, slope is of the tangent line at the point, (3,4). Recall the equation for a line is: Answer [This object is a pull tab] Slide 209 / 213 Example Slide 209 (Answer) / 213 Example Example: Find the slope of the graph of at the point Example: Find the slope of the graph of at the point Answer Plug in point values: 1. Differentiate both sides 2. Collect all dy/dx to one side 3. Factor out dy/dx 4. Solve for dy/dx. [This object is a pull tab]

61 Slide 210 / 213 Slide 210 (Answer) / 213 Slide 211 / Find the slope of the tangent line at x=3 for the equation: A B C D Slide 211 (Answer) / Find the slope of the tangent Note: Students line may at x=3 need prompting for the to equation: A B C D Answer substitute the x-value into original function to find the corresponding y-value to use in derivative. Answer: A note: [This object is a pull tab] Slide 212 / Find the slope of the tangent line at the point for the equation: A Slide 212 (Answer) / Find the slope of the tangent line at the point for the equation: Answer: D A Answer note: B C D B C D [This object is a pull tab]

62 Slide 213 / Find the equation of tangent line through point (1, -1) for the equation: A Slide 213 (Answer) / Find the equation of tangent line through point (1, -1) for the equation: Answer: D A Answer note: B C D B C D slope at (1,-1) = -4 [This object is a pull tab]