Calculus: Quarter Two

Calculus: Quarter Two#

Celebration of Knowledge#

  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. 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. 2015, Free Response, #6

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

  1. Write an equation for the line tangent to the curve at the point (-1, 1).

  2. Find the coordinates of all points on the curve at which the line tangent to the curve at that point is vertical.

  3. Evaluate \frac{d^2 y}{dx^2} at the point on the curve where x = -1 and y = 1.

  1. 2017, Free Response, #6

../../_images/2017_apcalc_frp_06.png

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.

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

  2. Let k be the function defined by k(x)=h(f(x)). Find k^{\prime}(\pi).

  3. Let m be the function defined by m(x) = g(-2x) \cdot h(x). Find m^{\prime}(2).

  1. 2015, Free Response, #5

../../_images/2015_apcalc_frp_05.png

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.

  1. Find all x-coordinates at which f has a relative maximum. Give a reason for your answer.

  2. 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.

  3. Find the x-coordinates of all points of inflection for the graph of f. Give a reason for your answer.

  1. 2023, Free Response, #6

Consider the curve given by the equation,

6xy = 2 + y^3

  1. Show that,

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

  1. 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.

  2. 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.

  3. 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?