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