Graphing#

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.

2021, Free Response, #5

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

Show that $\frac{dy}{dx}=\frac{y \cos{x}}{4y - \sin{x}}$ .
Write an equation for the line tangent to the curve at the point $(0, \sqrt{3})$ .
For $0 \leq x \leq \pi$ and $y > 0$ , find the coordinates of the point where the line tangent to the curve is horizontal.
Determine whether $f$ has a relative minimum, a relative maximum, or neither at the point found in part c. Justify your answer.

2005, Free Response, #4

x	0	$0 < x < 1$	1	$1 < x < 2$	2	$2 < x < 3$	3	$3 < x < 4$
$f(x)$	-1	Negative	0	Positive	2	Positive	0	Negative
$f^{\prime}(x)$	4	Positive	0	Positive	DNE	Negative	-3	Negative
$f^{\prime}{\prime}(x)$	-2	Negative	0	Positive	DNE	Negative	0	Positive

Let f be a function that is continuous on the interval $[0, 4)$ . The function f is twice differentiable except at $x = 2$ . The function f and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of f do not exist at x = 2.

For $0 < x < 4$ , find all values of x at which f has a relative extremum. Determine whether f has a relative maximum or a relative minimum at each of these values. Justify your answer.
Sketch the graph of a function that has all the characteristics of f .

2023, Free Response, #4

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

Does f have a relative minimum, a relative maximum, or neither at $x = 6$ ? Give a reason for your answer.
On what open intervals, if any, is the graph of f concave down? Give a reason for your answer.
Find the value of

$\lim_{x \to 2} \frac{6 \cdot f(x) - 3x}{x^2 - 5x + 6}$

or show that it does not exist. Justify your answer.

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

2022, Free Response, #3

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

Find $f(0)$ and $f(5)$ .
Find the $x$ -coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$ . Justify your answer.
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.
For the function $g$ defined in part c, find the absolute minimum value on the interval $0 \leq x \leq 7$ . Justify your answer.

2021, Free Response, #5

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

Show that

$\frac{dy}{dx} = \frac{y \cdot \cos(x)}{4y - \sin(x)}$

Write an equation for the line tangent to the curve at the point $(0, \sqrt{3})$
For $0 \leq x \leq \pi$ and $y \geq 0$ , find the coordinates of the point where the line tangent to the curve is horizontal.
Determine whether f has a relative minimum, a relative maximum, or neither at the point found in part c. Justify your answer.

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 $x = 2$ .

2006, Free Response, Form B, #3

../../../_images/2006_apcalc_frp_formb_03.png

The figure above is the graph of a function of x, which models the height of a skateboard ramp. The function meets the following requirements.

At $x = 0$ , the value of the function is 0, and the slope of the graph of the function is 0.

At $x = 4$ , the value of the function is 1, and the slope of the graph of the function is 1.

Between $x = 0$ and $x = 4$ , the function is increasing.

Let $f(x) = a x ^ 2$ , where $a$ is a nonzero constant. Show that it is not possible to find a value for a so that $f$ meets requirement ii above.
Let $g(x) = cx^3 - \frac{x^2}{16}$ , where $c$ is a nonzero constant. Find the value of $c$ so that g meets requirement ii above. Show the work that leads to your answer.
Using the function $g$ and your value of $c$ from part b, show that $g$ does not meet requirement iii above.
Let $h(x) = \frac{x^n}{k}$ , where $k$ is a nonzero constant and $n$ is a postive integer. Find the values of $k$ and $n$ so that $h$ meets requirement ii above. Show that h also meets requirements i and ii above.

2017, Free Response, #3

The function $f$ on the closed interval $[-6, 5]$ and satisfies $f(-2)=7$ . The graph of $f^{\prime}$ , the derivative of $f$ , consists of a semicircle and three line segments, as shown in the figure above.

Find the values of $f(-6)$ and $f(5)$ .
On what intervals is $f$ increasing? Justify your answer.
Find the absolue minimum value of $f$ on the closed interval $[-6, 5]$ . Justify your answer.
For each of $f^{\prime \prime}(-5)$ and $f^{\prime}{\prime}(3)$ , find the value or explain why it does not exist.

2018, Free Response, #5

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

Find the average rate of change of $f$ on the interval $0 \leq x \leq \pi$ .
What is the slope of the line tangent to the graph at $x = \frac{3 \pi}{2}$ ?
Find the absolute minimum value of $f$ on the interval $0 \leq x \leq 2 \pi$ . Justify your answer.
Let $g$ be a differentiable function such that $g(\frac{\pi}{2})=0$ . The graph of $g^{\prime}$ , the derivatibve of $g$ , is shown below. Find the value of $\lim_{x \to \frac{\pi}{2}} \frac{f(x)}{g(x)}$ or state that it does not exist. Justify your answer.