Unit 3 Applications of Differentiation Lesson 4: The First Derivative Lesson 5: Concavity and The Second Derivative

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1 Warmup 1) The lengths of the sides of a square are decreasing at a constant rate of 4 ft./min. In terms of the perimeter, P, what is the rate of change of the area of the square in square feet per minute? (a) -2 P (b) 2 P (c) -4 P (d) 4 P (e) 8 P 2) The base of a triangle is decreasing at a constant rate of 0.2 cm/sec and the height is increasing at 0.1 cm/sec. If the area is increasing, which answer best describes the constraints on the height h at the instant when the base is 3 centimeters? (a) h > 3 (b) h <1 (c) h > 1.5 (d) h < 1.5 (e) h > 2 3) Let y f (x) be a differentiable function on [-10, 3]. If f ( 10) 5, f ( 1) 2, and f (3) 5 : a) What is the minimum number of zeros that this function could have? b) Is there a value of x for which f ( x) 2? c) Does f have any horizontal tangents? d) If f has only one extremum on [-10, 3], will it be a maximum or a minimum?

2 How Do Functions Speak to Us? Take a look at the graph of f ( x) below. What kinds of things can we conclude about the graph of f( x )? Summarizing Increasing and Decreasing If f ( x) is, then f( x) is on that interval. If f ( x) is, then f( x) is on that interval. If f ( x) is, then f( x) is on that interval.

3 Ex 1: Find all intervals where 1 f ( x) x sin x is increasing and decreasing on [0, 2 ]. 2 The First Derivative Test If a function changes from to on an interval, then there is a at the critical value. If a function changes from to on an interval, then there is a at the critical value. Ex 2: extrema. Find all intervals where f ( x) ( x 4) is increasing or decreasing. Also, find any relative

4 A Classic AP Derivative Problem Let f be the function given by 3 f ( x) x 7x 6. (a) Find the zeros of f. (b) Write the equation of the line tangent to the graph of f at x 1. (c) Find the local maximum and minimum values for the function. (d) Find the number c that satisfies the conclusion of the Mean Value Theorem for f on the interval [1, 3].

5 Summarizing Concavity Where f ( x) 0 the graph of f is curving downward. This is called. Where f ( x) 0 the graph of f is curving upward. This is called. Ex 1: Determine the intervals where the graph of f ( x) 6( 3) 2 1 x is concave up or concave down. Points of Inflection A point of inflection is a point on a graph where the concavity changes. The EXCEPTION: Points of inflection do not exist, if the graph changes concavity because of. Why?

6 Ex 2: Identify all intervals where the function 3 f ( x) x ( x 4) is increasing or decreasing. Find all points of extrema. Also, determine where the graph is concave up and concave down. Finally, label all points of inflection. The Second Derivative Test If f ( c) 0, then we know that f( x ) must have a at c. Additionally, if f ( c) 0, then we also know that f( x) must be. Therefore, what type of extrema must this indicate? A Relative If f ( c) 0, then we know that f( x ) must have a at c. Additionally, if f ( c) 0, then we also know that f( x) must be. Therefore, what type of extrema must this indicate? A Relative

7 Mystery Challenge Sketch a graph of f( x ) based on the following information. f(0) f(2) 0 f ( x) 0 if x 1 f (1) 0 f ( x) 0 if x 1 f ( x) 0 Another Classic AP Derivative Problem Let f be a function which is twice differentiable for all real numbers and which satisfies the following properties. I. f (0) 1 II. f ( x) 0 for all x 0 III. f is concave down for all x 0 and f is concave up for all x 0. Let (a) Sketch a possible graph for f which takes into account its properties given above. g x 2 ( ) f ( x ) (b) Find the x-coordinate of all relative minimum point(s) of g( x ). Justify your answer.

8 Another Classic AP Derivative Problem (continued) (c) Where is the graph of g( x) concave up? Justify your answer. (d) Use the information obtained in the previous three parts to sketch a possible graph of g( x ).