AP®MathsCollege BoardCalculus ABExam QuestionsUnit 5: Analytical Applications of DifferentiationUnit 5 Overview

Unit 5 Overview (College Board AP® Calculus AB): Exam Questions

36 mins4 questions
1a
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1 mark

The function f is defined on the closed interval [-2, 8] and satisfies f(2) = 1. The graph of f', the derivative of f, consists of two line segments and a semicircle, as shown in the figure.

Graph of f′: piecewise curve on x from −2 to 8 with V-shape through (0,−2) to (4,2), then a semicircle from (4,2) down to (6,0) and up to (8,2).

Does f have a relative minimum, a relative maximum, or neither at x = 6? Give a reason for your answer.

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1b
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2 marks

On what open intervals, if any, is the graph of f concave down? Give a reason for your answer.

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1c
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3 marks

Find the value of \displaystyle \lim_{x \to 2} \frac{6 f(x) - 3 x}{x^2 - 5 x + 6}, or show that it does not exist. Justify your answer.

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1d
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3 marks

Find the absolute minimum value of f on the closed interval [-2, 8]. Justify your answer.

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2a
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2 marks

Consider the function y = f(x) whose curve is given by the equation 2y^2 - 6 = y\sin x for y > 0.

Show that \dfrac{\text{d}y}{\text{d}x} = \dfrac{y\cos x}{4y - \sin x}.

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2b
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1 mark

Write an equation for the line tangent to the curve at the point (0, \sqrt{3}).

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2c
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3 marks

For 0 \leq x \leq \pi and y > 0, find the coordinates of the point where the line tangent to the curve is horizontal.

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2d
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3 marks

Determine whether f has a relative minimum, a relative maximum, or neither at the point found in part (c). Justify your answer.

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3a
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1 mark

Let f be the function defined by f(x) = e^x\,\text{cos} \; x.

Find the average rate of change of f on the interval 0 \leq x \leq \pi.

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3b
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2 marks

What is the slope of the line tangent to the graph of f at x = \dfrac{3\pi}{2}?

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3c
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3 marks

Find the absolute minimum value of f on the interval 0 \leq x \leq 2\pi. Justify your answer.

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3d
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3 marks

Let g be a differentiable function such that g blank space open parentheses pi over 2 close parentheses equals 0. The graph of g', the derivative of g, is shown below.

Graph of g′: curve rises from (0,−0.5) to (π/2,2), then straight line falls crossing x-axis near 3π/2 and ending below at (2π,−0.5).

Find the value of \displaystyle\lim_{x \to \pi/2} \dfrac{f(x)}{g(x)}, or state that it does not exist. Justify your answer.

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4a
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2 marks

Let f be the function defined by f(x) = \text{cos} \; (2x) + e^{\text{sin} \; x}. Let g be a differentiable function. The table below gives values of g and its derivative g' at selected values of x.

x

g open parentheses x close parentheses

g apostrophe open parentheses x close parentheses

-5

10

-3

-4

5

-1

-3

2

4

-2

3

1

-1

1

-2

0

0

-3

Let h be the function whose graph, consisting of five line segments, is shown below.

Piecewise linear graph of function h on x–y axes, rising, flat, falling through origin, then dipping below x-axis before sharply rising again on the right.

Find the slope of the line tangent to the graph of f at x = \pi.

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4b
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2 marks

Let k be the function defined by k(x) = h(f(x)). Find k'(\pi).

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4c
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3 marks

Let m be the function defined by m(x) = g(-2x)\,h(x). Find m'(2).

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4d
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2 marks

Is there a number c in the closed interval [-5, -3] such that g'(c) = -4? Justify your answer.

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