AP Calculus BC. Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines

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1 AP Calculus BC Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines

2 Essential Questions & Why: Essential Questions: What is the difference between average and instantaneous rates of change and how can I calculate them? What are tangent and normal lines to a curve and how can I write equations for them? Why? Instantaneous rates of change and tangent lines are common applications of the derivative, our next big idea in our study of the calculus.

3 Learning Targets: Apply the definition of slope of a curve in order to calculate slopes. Write equations of the tangent line and normal line to a curve at a given point. Calculate the average rate of change of a function over a specified interval.

4 Learning Objective: Big Idea 1: Limits Enduring Understanding 1.2: Students will understand that continuity is a key property of functions that is defined using limits. Learning Objective 1.2A: Students will be able to analyze functions for intervals of continuity or points of discontinuity. Essential Knowledge 1.2A1: Students will know that a function ƒ is continuous at x = c provided that ƒ(c) exists, lim x c f x ( ) exists, and lim x c f x ( ) = f c ( ).

5 Learning Objective: Big Idea 1: Limits EU 1.2: Continuity is a key property of functions that is defined using limits. LO 1.2A: Analyze functions for intervals of continuity or points of discontinuity. EK 1.2A2: Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous at all points in their domains. EK 1.2A3: Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

6 Learning Objective: Big Idea 1: Limits Enduring Understanding 1.2: Students will understand that continuity is a key property of functions that is defined using limits. Learning Objective 1.2B: Students will be able to determine the applicability of important calculus theorems using continuity. Essential Knowledge 1.2B1: Students will know that continuity is an essential condition for theorems such as the Intermediate Value Theorem.

7 Mathematical Practices for AP Calculus MPAC 2: Connecting concepts. Students can relate the concept of a limit to all aspects of calculus.

8 Quote for today: I know that you believe that you understood what you think I said, but I am not sure you realize that what you heard is not what I meant. Robert McCloskey ( )

9 Average Rate of Change: The average rate of change of a quantity over a period of time is the amount of change divided by the time it takes. In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval.

10 Slope of a Curve at a Point: The slope of the curve y = f(x) at the point P(a, f(a)) is the number m below, provided the limit exists: m = lim h 0 f a + h ( ) f a ( ) h The tangent line to the curve at P is the line through P with this slope.

11 Difference Quotient: The expression f a + h ( ) f a ( ) is the difference quotient of ƒ at a. Suppose the difference quotient has a limit as h approaches 0. If we interpret the difference quotient as a secant slope, the limit is the slope of both the curve and the tangent to the curve at the point x = a. h

12 Difference Quotient: If we interpret the difference quotient as an average rate of change, the limit is the function s rate of change with respect to x at the point x = a.

13 Normal to a Curve: The normal line to a curve at a point is the line perpendicular to the tangent at that point.

14 Assignments: CW 2.4: #3, 5, 7, 11, 13, 17, 23, 25, 27, & 29. HW 2.4: Get caught up. Ch. Two Test: Wednesday, October 4.