AP®MathsCollege BoardCalculus ABExam QuestionsUnit 6: Integration & Accumulation of ChangeUnit 6 Overview

Unit 6 Overview (College Board AP® Calculus AB): Exam Questions

2 hours11 questions

3 marks

Graph of f′: semicircle from (0,0) to (4,0) dipping to y = −2, then line up to (6,2) and line down to (7,1) on x–y axes.

Let $f$ be a differentiable function with $f (4) = 3$ . On the interval $0 \leq x \leq 7$ , the graph of $f'$ , the derivative of $f$ , consists of a semicircle and two line segments, as shown in the figure above.

Find $f (0)$ and $f (5)$ .

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2 marks

Find the $x$ -coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$ . Justify your answer.

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2 marks

Let $g$ be the function defined by $g (x) = f (x) - x$ . On what intervals, if any, is $g$ decreasing for $0 \leq x \leq 7$ ? Show the analysis that leads to your answer.

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2 marks

For the function $g$ defined in part (c), find the absolute minimum value on the interval $0 \leq x \leq 7$ . Justify your answer.

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2 marks

An ice sculpture melts in such a way that it can be modelled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function $r$ , where $r (t)$ is measured in centimeters and $t$ is measured in days. The table below gives selected values of $r' (t)$ , the rate of change of the radius, over the time interval $0 \leq t \leq 12$ .

$t$ (days)	0	3	7	10	12
$r' (t)$ (cm per day)	$- 6.1$	$- 5.0$	$- 4.4$	$- 3.8$	$- 3.5$

Approximate $r'' (8.5)$ using the average rate of change of $r'$ over the interval $7 \leq t \leq 10$ . Show the computations that lead to your answer, and indicate units of measure.

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2 marks

Is there a time $t$ , $0 \leq t \leq 3$ , for which $r' (t) = - 6$ ? Justify your answer.

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2 marks

Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of $\int_{0}^{12} r' (t) d t$ .

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3 marks

The height of the cone decreases at a rate of 2 centimetres per day. At time $t = 3$ days, the radius is 100 centimetres and the height is 50 centimetres. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time $t = 3$ days. (The volume $V$ of a cone with radius $r$ and height $h$ is $V = \frac{1}{3} π r^{2} h$ .)

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10a

2 marks

The functions $f$ and $g$ are twice differentiable. The table gives values of the functions and their first derivatives at selected values of $x$ .

$x$	$0$	$2$	$4$	$7$
$f (x)$	$10$	$7$	$4$	$5$
$f' (x)$	$\frac{3}{2}$	$- 8$	$3$	$6$
$g (x)$	$1$	$2$	$- 3$	$0$
$g' (x)$	$5$	$4$	$2$	$8$

Let $h$ be the function defined by $h (x) = f (g (x))$ . Find $h' (7)$ . Show the work that leads to your answer.

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10b

3 marks

Let $k$ be a differentiable function such that $k' (x) = (f (x))^{2} \cdot g (x)$ . Is the graph of $k$ concave up or concave down at the point where $x = 4$ ? Give a reason for your answer.

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10c

1 mark

Let $m$ be the function defined by $m (x) = 5 x^{3} + \int_{0}^{x} f' (t) d t$ . Find $m (2)$ . Show the work that leads to your answer.

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10d

3 marks

Is the function $m$ defined in part (c) increasing, decreasing, or neither at $x = 2$ ? Justify your answer.

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11a

3 marks

The functions $f$ and $g$ have continuous second derivatives. The table below gives values of the functions and their derivatives at selected values of $x$ .

$x$	$f (x)$	$f' (x)$	$g (x)$	$g' (x)$
1	-6	3	2	8
2	2	-2	-3	0
3	8	7	6	2
6	4	5	3	-1

Let $k (x) = f (g (x))$ . Write an equation for the line tangent to the graph of $k$ at $x = 3$ .

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11b

3 marks

Let $h (x) = \frac{g (x)}{f (x)}$ . Find $h' (1)$ .

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11c

3 marks

Evaluate $\int_{1}^{3} f'' (2 x) d x$ .

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