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 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 "
Slide 5 (Answer) / 213 Slide 6 / 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 Teacher Notes Exploration into the idea of being locally linear... 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 Rate of Change Click here to go to the lab titled "Derivatives allow them to be able to find the slopes of curves in particular places, due to the fact Exploration: y = x 2 " that they are locally linear. URL for Lab: http://njctl.org/courses/math/ Return to ap-calculus-ab/derivatives/x-squared- Table of exploration-lab/ [This object is a teacher notes pull tab] Contents Slide 7 / 213 Slide 7 (Answer) / 213 Road Trip! Road Trip! Consider the following scenario: Consider the following scenario: You and your friends take a road trip and leave You and your friends take a road trip and leave When students answer 60 mph, ask the at 1:00pm, drive 240 miles, and arrive at 5:00pm. at 1:00pm, drive 240 miles, and arrive at 5:00pm. following: How fast were you driving? How fast were you driving? Teacher Notes · 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 Position vs. Time Position vs. Time Students can often grasp the concept of Now, consider the following position vs. time graph: Now, consider the following position vs. time graph: Teacher Notes derivatives when you relate it to something they are familiar with, such as velocity. Discuss with students: · What does the orange line represent? position position (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 ) t 1 t 2 t 3 t 1 t 2 t 3 time time t 0 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 11 (Answer) / 213 Slope of a Secant Line How would you calculate the slope of the secant line? y 2 b a y 1 x 1 x 2 Slide 12 / 213 Slide 12 (Answer) / 213 Slope of a Secant Line Slope of a Secant Line Allow students to discuss what they think, What happens to the slope of the secant line as the point b What happens to the slope of the secant line as the point b eventually listening for the conclusion that the moves closer to the point a? moves closer to the point a? secant line resembles the tangent line as Teacher Notes those points get closer together. b b y 2 y 2 Encourage them to observe the fact that the change in x, # x, gets smaller (approaching 0) as the point b approaches a. a a y 1 y 1 In reference to the second question, students x 1 x 1 x 2 x 2 should note that when b=a, using 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?
Slide 13 / 213 Slide 13 (Answer) / 213 Slide 14 / 213 Slide 14 (Answer) / 213 Slide 15 / 213 Slide 15 (Answer) / 213
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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)"
Slide 24 / 213 Slide 25 / 213 Slide 25 (Answer) / 213 Slide 26 / 213 Slide 26 (Answer) / 213 Slide 27 / 213
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.
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
Slide 35 (Answer) / 213 Slide 36 / 213 The Constant Rule Teacher Notes 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, Why do you think this is? they will conclude that they are all equations of horizontal lines. Think of the meaning of a derivative, Therefore, no matter where you are and how it applies to the graph of on the graph, the slope of any [This object is a teacher notes pull tab] each of these functions. tangent line will be zero. Hence, the derivative is zero at any point, regardless of the x-value. where c is a constant Slide 37 / 213 Slide 37 (Answer) / 213 Slide 38 / 213 Slide 38 (Answer) / 213
Slide 39 / 213 Slide 39 (Answer) / 213 Slide 40 / 213 Slide 40 (Answer) / 213 11 11 A A Answer B B C C C D D [This object is a pull tab] E E Slide 41 / 213 Slide 41 (Answer) / 213 12 What is the derivative of 15? 12 What is the derivative of 15? Answer A A x x D B B 1 1 C C 14 14 [This object is a pull tab] D D 0 0 E E -15 -15
Slide 42 / 213 Slide 42 (Answer) / 213 Slide 43 / 213 Slide 43 (Answer) / 213 14 Find y' if 14 Find y' if Answer A C A C C [This object is a pull tab] B D B D Slide 44 / 213 Slide 44 (Answer) / 213
Slide 45 / 213 Slide 45 (Answer) / 213 Slide 46 / 213 Slide 46 (Answer) / 213 Slide 47 / 213 Slide 47 (Answer) / 213
Slide 48 / 213 Slide 48 (Answer) / 213 Slide 49 / 213 Slide 49 (Answer) / 213 19 Find y'(16) if 19 Find y'(16) if Answer A C A C B E E [This object is a pull tab] B D B D Slide 50 / 213 Slide 51 / 213 Higher Order Derivatives You may be wondering.... Can you find the derivative of a derivative!!?? The answer is... YES! Higher Order Derivatives 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. Return to Table of Contents
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