Graphing

Graphing#

  1. 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.

  1. Let h be the function defined by h(x)=f(g(x)). Find h^{\prime}(7). Show the work that leads to your answer.

  2. 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.

  1. 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.

  1. Show that \frac{dy}{dx}=\frac{y \cos{x}}{4y - \sin{x}}.

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

  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.

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

  1. 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.

  1. 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.

  2. Sketch the graph of a function that has all the characteristics of f .

  1. 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,

../../../_images/2023_apcalc_frp_04.png
  1. Does f have a relative minimum, a relative maximum, or neither at x = 6? Give a reason for your answer.

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

  3. 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.

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

  1. 2022, Free Response, #3

../../../_images/2022_apcalc_frp_03.png

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.

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

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

  3. 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.

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

  1. 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.

  1. Show that

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

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

  2. 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.

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

  1. 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.

  1. 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.

  2. On the interval 0 \leq x \leq 3, find the value of x at which f has an absolute maximum. Justify your answer.

  3. Write an equation for the line tangent to the graph of f at x = 2.

  1. 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.

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

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

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

  1. 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.

  2. 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.

  3. Using the function g and your value of c from part b, show that g does not meet requirement iii above.

  4. 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.

  1. 2017, Free Response, #3

../../../_images/2017_apcalc_frp_03.png

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.

  1. Find the values of f(-6) and f(5).

  2. On what intervals is f increasing? Justify your answer.

  3. Find the absolue minimum value of f on the closed interval [-6, 5]. Justify your answer.

  4. For each of f^{\prime \prime}(-5) and f^{\prime}{\prime}(3), find the value or explain why it does not exist.

  1. 2018, Free Response, #5

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

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

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

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

  4. 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.

../../../_images/2018_apcalc_frp_05.png