Calculus: Quarter Two#

Celebration of Knowledge#

2006, Free Response, Form B, #2

../../../_images/2006_apcalc_frp_formb_02.png

Let $f`$ be the function defined for $x \geq 0$ with $f (0) = 5$ and $f^{\prime}$ , the first derivative of $f$ , given by $f^{\prime}( x ) = e ^{ - \frac{x}{4} } \cdot \sin{x^2}$ . The graph of $y = f^{\prime}( x )$ is shown above.

Use the graph of $f^{\prime}$ to determine whether the graph of $f$ is concave up, concave down, or neither on the interval $1.7 < x < 1.9$ . Explain your reasoning.
On the interval $0 \leq x \leq 3$ , find the value of x at which f has an absolute maximum. Justify your answer.
Write an equation for the line tangent to the graph of $f$ at $x = 2$ .

2023, Free Response, #5

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

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

Let $h$ be the function defined by $h(x)=f(g(x))$ . Find $h^{\prime}(7)$ . Show the work that leads to your answer.
Let $k$ be a differentiable function such that $k^{\prime}=(f(x))^2 \cdot g(x)$ . Is the graph of $k$ concave up or down at the point where $x = 4$ ? Give a reason for your answer.

2015, Free Response, #6

Consider the curve given by the equation $y^3 - xy = 2$ .

Write an equation for the line tangent to the curve at the point $(-1, 1)$ .
Find the coordinates of all points on the curve at which the line tangent to the curve at that point is vertical.
Evaluate $\frac{d^2 y}{dx^2}$ at the point on the curve where $x = -1$ and $y = 1$ .

2017, Free Response, #6

x	$g(x)$	$g^{\prime}(x)$
-5	10	-3
-4	5	-1
-3	2	4
-2	3	1
-1	1	-2
0	0	-3

Let $f$ be the function defined by $f(x)=\cos{2x} + e^{\sin(x)}$ .

Let $g$ be a differentiable function. The table above gives values of $g$ and its derivative $g^{\prime}$ at selected values of $x$ .

Let $h$ be the function whose graph, consisting of five line segments, is shown in the figure above.

Find the slope of the line tangent to the graph of $f$ at $x = \pi$ .
Let $k$ be the function defined by $k(x)=h(f(x))$ . Find $k^{\prime}(\pi)$ .
Let $m$ be the function defined by $m(x) = g(-2x) \cdot h(x)$ . Find $m^{\prime}(2)$ .

2015, Free Response, #5

The figure above shows the graph of $f^{\prime}$ , the derivative of a twice-differentiable function $f$ , on the interval $[-3,4]$ . The graph of $f^{\prime}$ has horizontal tangents at $x=-1$ , $x=1$ , and $x=3$ .

Find all x-coordinates at which f has a relative maximum. Give a reason for your answer.
On what open intervals contained in $-3 < x < 4$ is the graph of f both concave down and decreasing? Give a reason for your answer.
Find the x-coordinates of all points of inflection for the graph of f. Give a reason for your answer.

2023, Free Response, #6

Consider the curve given by the equation,

$6xy = 2 + y^3$

Show that,

$\frac{dy}{dx} = \frac{2y}{y^2 - 2x}$

Find the coordinates of a point on the curve at which the line tangent to the curve is horizontal, or explain why no such point exists.
Find the coordinates of a point on the curve at which the line tangent to the curve is vertical, or explain why no such point exists.
A particle is moving along the curve. At the instance when the particle is at the point $(\frac{1}{2}, -2)$ , its horizontal position is increasing at a rate of $\frac{dx}{dt}=\frac{2}{3}$ units per second. What is the value of $\frac{dy}{dt}$ , the rate of change of the particle’s vertical position, at that instant?

Calculus: Quarter Two

Contents

Calculus: Quarter Two#

Celebration of Knowledge#