Home / AP Calculus AB: 5.8 Sketching Graphs of Functions and Their  Derivatives – Exam Style questions with Answer- MCQ

AP Calculus AB: 5.8 Sketching Graphs of Functions and Their  Derivatives – Exam Style questions with Answer- MCQ

AP Calculus AB-Questions and Answers - All Topics

Question

The table above gives values of a function \( f \) at selected values of \( x \). If \( f \) is twice-differentiable on the interval \( 1 \leq x \leq 5 \), which of the following statements could be true?

\( x \)12345
\( f(x) \)940-3-5

A) \( f’ \) is negative and decreasing for \( 1 \leq x \leq 5 \)
B) \( f’ \) is negative and increasing for \( 1 \leq x \leq 5 \)
C) \( f’ \) is positive and decreasing for \( 1 \leq x \leq 5 \)
D) \( f’ \) is positive and increasing for \( 1 \leq x \leq 5 \)

▶️ Answer/Explanation

Solution

Approximate \( f’ \) using differences: \( f'(x) \approx \frac{f(x+1) – f(x)}{1} \).
\( x = 1 \): \( f'(1) \approx 4 – 9 = -5 \) (negative).
\( x = 2 \): \( f'(2) \approx 0 – 4 = -4 \) (negative).
\( x = 3 \): \( f'(3) \approx -3 – 0 = -3 \) (negative).
\( x = 4 \): \( f'(4) \approx -5 – (-3) = -2 \) (negative).
\( f’ \) is negative. Check \( f” \) (change in \( f’ \)): \( -4 – (-5) = 1 \), \( -3 – (-4) = 1 \), \( -2 – (-3) = 1 \) (positive), so \( f’ \) is increasing.
Answer: B
Final Answer: \( f’ \) is negative and increasing for \( 1 \leq x \leq 5 \)

Question

The graph of the function f is shown above for \(-2\leq x\leq 2\) . Which of the following could be the graph of an antiderivative of f ?

A

B

C

D

▶️Answer/Explanation

Ans:D

Question

A symmetric power function graph with an increasing slope is plotted on a coordinate plane. It starts from the third quadrant and rises upward to enter the second quadrant and then entering the first quadrant cutting the positive y axis. It appears rising upward as it enters the first quadrant.

The graph of y=h(x) is shown above. Which of the following could be the graph of \(y=h{}'(x)\)?

A A symmetric power function graph with a decreasing slope is plotted on a coordinate plane. It starts from the second quadrant and slopes downward to enter the first quadrant, cutting the positive y axis. It then enters the fourth quadrant cutting the positive axis. It appears sloping downward as it enters the fourth quadrant.

B A graph starts from third quadrant and intersects the negative x axis at a point. It further increases in the shape of a root curve and intersects positive y axis at a point, and finally terminates in the first quadrant.

CA parabola that opens downward is plotted on a coordinate plane. It has its left x-intercept lying in the third quadrant and right x-intercept lying in the fourth quadrant. The vertex of the parabola lies at a point on the positive y axis.

DA coordinate plane shows an upward opening parabola having vertex at negative y axis. The ends of the parabola are slightly curved and fall in first and third quadrants, respectively.

E A parabola that opens upward is plotted on a coordinate plane. It has its left x-intercept lying in the second quadrant and right x-intercept lying in the first quadrant. The vertex of the parabola lies at a point on the negative y axis.

▶️Answer/Explanation

Ans:E

Question

A line curve of y equals f of x is plotted on the xy plane. The curve is an inverted symmetric curve plotted with the vertex lying on the x axis.

The graph of y = f(x) on the closed interval [0, 4] is shown above. Which of the following could be the graph y = f'(x)?

AA bell-shaped curve is plotted on the xy plane.

B A sinusoidal curve is plotted on the xy plane.

CA line graph plotted on the xy plane, starting from negative y axis rising upward cutting the positive x axis to enter the first quadrant.

DA line graph plotted on the xy plane, starting from the origin sloping downward and then rising upward to enter the first quadrant cutting the positive x axis and then slopes downward further entering the fourth quadrant cutting the positive x axis.

E A line graph plotted on the xy plane, starting from the origin in the fourth quadrant and then rising upward to the first quadrant. It further slopes downward again entering into the fourth quadrant.

▶️Answer/Explanation

Ans:D

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