Slide 1 / 213 Slide 2 / 213 AP Calculus Derivatives 2015-11-03 www.njctl.org Slide 3 / 213 Slide 4 / 213 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 Slide 5 / 213 Slide 6 / 213 Derivatives Exploration Exploration into the idea of being locally linear... Rate of Change Click here to go to the lab titled "Derivatives Exploration: y = x 2 " Return to Table of Contents
Slide 7 / 213 Slide 8 / 213 Position vs. Time Road Trip! Now, consider the following position vs. time graph: 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? position t 1 t 2 t 3 time t 0 Slide 9 / 213 Slide 10 / 213 SECANT vs. TANGENT Recap A secant line connects 2 points on a curve. The slope b y 2 of this line is also known as We will discuss more about average and instantaneous the Average Rate of Change. velocity in the next unit, but hopefully it allowed you to a see the difference in calculating slopes at a specific point, y 1 A tangent line touches one rather than over a period of time. point on a curve and is x 1 x 2 known as the Instantaneous Rate of Change. Slide 11 / 213 Slide 12 / 213 Slope of a Secant Line Slope of a Secant Line What happens to the slope of the secant line as the point b moves closer to the point a? How would you calculate the slope of the secant line? b y 2 y 2 b a y 1 a y 1 x 1 x 2 x 1 x 2 What is the problem with the traditional slope formula when b=a?
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Slide 19 / 213 Slide 20 / 213 Slope of a Curve (Instantaneous Rate of Change) Return to Table of Contents Slide 21 / 213 Slide 22 / 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 23 / 213 Slide 24 / 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)"
Slide 25 / 213 Slide 26 / 213 Slide 27 / 213 Slide 28 / 213 Slide 29 / 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.
Slide 31 / 213 Slide 32 / 213 Derivative Rules: Power, Constant & Sum/Difference Return to Table of Contents Slide 33 / 213 Slide 34 / 213 Slide 35 / 213 Slide 36 / 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
Slide 37 / 213 Slide 38 / 213 Slide 39 / 213 Slide 40 / 213 11 A B C D E Slide 41 / 213 Slide 42 / 213 12 What is the derivative of 15? A x B 1 C 14 D 0 E -15
Slide 43 / 213 Slide 44 / 213 14 Find y' if C A B D Slide 45 / 213 Slide 46 / 213 Slide 47 / 213 Slide 48 / 213
Slide 49 / 213 Slide 50 / 213 19 Find y'(16) if C A Higher Order Derivatives E B D Return to Table of Contents Slide 51 / 213 Slide 52 / 213 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. Slide 53 / 213 Slide 54 / 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 55 / 213 Slide 56 / 213 Slide 57 / 213 Slide 58 / 213 22 Find if A B C D E Slide 59 / 213 Slide 60 / 213
Slide 61 / 213 Slide 62 / 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! Derivatives of Trig Functions For example, if asked to take the derivative of , our previous rules Return to would not apply. Table of Contents Slide 63 / 213 Slide 64 / 213 Proof Let's take a moment to prove one of these derivatives... Slide 65 / 213 Slide 66 / 213
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Slide 73 / 213 Slide 74 / 213 34 Find A D B E C F Slide 75 / 213 Slide 76 / 213 36 Find A D B E C F Slide 77 / 213 Slide 78 / 213 Derivative Rules: Product & Quotient Return to Table of Contents
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Slide 91 / 213 Slide 92 / 213 44 What About Rational Functions? So far, we have discussed how to take the derivatives of True polynomials using the Power Rule, Sum and Difference Rule, and False 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. Slide 93 / 213 Slide 94 / 213 Example Given: Find f(x), or "top" g(x), or "bottom" Slide 95 / 213 Slide 96 / 213 Example Proof Now that you have seen the Quotient Rule in action, we can revisit Given: Find one of the trig derivatives and walk through the proof.
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Slide 103 / 213 Slide 104 / 213 Derivatives Using Tables On the AP Exam, in addition to calculating derivatives on your own, you must also be able Calculating Derivatives to use tabular data to find derivatives. These problems are not incredibly difficult, but can be Using Tables distracting due to extraneous information. Return to Table of Contents Slide 105 / 213 Slide 106 / 213 Slide 107 / 213 Slide 108 / 213
Slide 109 / 213 Slide 110 / 213 Slide 111 / 213 Slide 112 / 213 Slide 113 / 213 Slide 114 / 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 Equations of Tangent & particular point, therefore we will need a slope at that specific point. Normal Lines Now that we are familiar with calculating derivatives (slopes) we can use our techniques to write these equations of tangent lines. Return to Table of Contents
Slide 115 / 213 Slide 116 / 213 Slide 117 / 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 tangent line normal line at x = 1 at x = 1 How do you suppose we would calculate the slope of a normal line? Slide 119 / 213 Slide 120 / 213
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Slide 127 / 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. Slide 131 / 213 Slide 132 / 213 Derivatives of Exponential Functions 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 cool! is the only nontrivial function whose derivative is the same as the function!
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