Home / AP Calculus AB 2.3 Estimating Derivatives of a Function at a Point – MCQs

AP Calculus AB 2.3 Estimating Derivatives of a Function at a Point - MCQs - Exam Style Questions

AP Calculus AB-MCQs and Answers - All Topics

No-Calc Question

The table below gives values of \(f\), \(f’\), \(g\), and \(g’\) for selected values of \(x\). If \(h(x)=f(g(x))\), what is the value of \(h'(1)\)?

\(x\)\(f(x)\)\(f'(x)\)\(g(x)\)\(g'(x)\)
\(-2\)\(-6\)\(9\)\(-10\)\(16\)
\(1\)\(5\)\(-3\)\(3\)\(-2\)
\(3\)\(0\)\(7\)\(8\)\(3\)

(A) \(-19\)   

(B) \(-14\)   

(C) \(7\)   

(D) \(9\)

▶️ Answer/Explanation

Chain rule: \(h'(x)=f'(g(x))\times g'(x)\).
At \(x=1\): \(g(1)=3\Rightarrow f'(g(1))=f'(3)=7\); also \(g'(1)=-2\).
So \(h'(1)=7\times(-2)=-14\).
✅ Answer: (B) \(-14\)

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.

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