Home / AP Calculus AB : 2.3 Estimating Derivatives  of a Function at a Point- Exam Style questions with Answer- MCQ

AP Calculus AB : 2.3 Estimating Derivatives  of a Function at a Point- Exam Style questions with Answer- MCQ

AP Calculus AB-Questions and Answers - All Topics

Question

Table of values for f(x)

Let \( f \) be the function given by \( f(x) = 2^{x^{2}} \). Selected values of \( f \) are given in the table above. If the values in the table are used to approximate \( f'(0.5) \), what is the difference between the approximation and the actual value of \( f'(0.5) \)?

A) 0
B) 0.176
C) 0.824
D) 1

▶️ Answer/Explanation

Solution

Correct Answer: B

The numerical value of the derivative at \( x = 0.5 \) obtained from the calculator is \( f'(0.5) = 0.824 \). A difference quotient can be used with the values in the table to estimate the derivative as: \[ \frac{f(1) – f(0)}{1 – 0} = \frac{2 – 1}{1} = 1 \] The error between the actual derivative value and this approximation is \( 0.176 \).

Question

Let f be the function given by \(f(x)=\frac{1}{7}x^{7}-\frac{7}{6}x^{6}+3x^{5}-\frac{5}{4}x^{4}-\frac{16}{3}x^{3}+6x^{2}\). Which of the following statements is true?

A f′(−1.1)<f′(0.5)<f′(1.4)

B f′(−1.1)<f′(1.4)<f′(0.5)

C f′(0.5)<f′(1.4)<f′(−1.1)

D f′(1.4)<f′(0.5)<f′(−1.1)

▶️Answer/Explanation

Ans:D
The calculator is used to store the expression for f(x) and to find the numerical derivative values at the three values of x. They are, in increasing order, f′(1.4)=−0.774, f′(0.5)=2.109, and f′(−1.1)=9.102.

Question

The graph of the trigonometric function f is shown above for a≤x≤b. At which of the following points on the graph of f could the instantaneous rate of change of f equal the average rate of change of f on the interval [a,b] ?

A A
B B
C C
D D

▶️Answer/Explanation

Ans:B
The instantaneous rate of change of f at the point B is the slope of the line tangent to the graph of f at the point B. The average rate of change of f on the interval [a,b] is the slope of the secant line through the points (a,f(a)) and (b,f(b)). The tangent line at B appears to be parallel to the secant line. Therefore, the instantaneous rate of change at B could be equal to the average rate of change.

Question

If  \( f(x)=2+|x-3|\)for all x, then the value of the derivative f ′(x )  at x = 3 is.

(A) -1                             (B) 0                                   (C) 1                    (D) 2                                     (E) nonexistent

▶️Answer/Explanation

Ans:E

 E There is no derivative at the vertex which is located at x = 3.

Question

The point on the curve \(x^{2}+2y=0\) that is nearest the point \( \left ( 0,-\frac{1}{2} \right )\) occurs where y is

(A) \( \frac{1}{2}\)                 (B) 0                        (C)\( -\frac{1}{2}\)                         (D) −1                       (E) none of the above

▶️Answer/Explanation

Ans:B

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