AP Calculus BC 3.3 Differentiating Inverse Functions- MCQs - Exam Style Questions
No-Calc Question
Let \(f(x)=x^{3}+x\). If \(g\) is the inverse of \(f\), what is the slope of the line tangent to the graph of \(g\) at \((2,1)\)?
(A) \(-\dfrac{1}{4}\)
(B) \(-\dfrac{1}{13}\)
(C) \(\dfrac{1}{13}\)
(D) \(\dfrac{1}{4}\)
(B) \(-\dfrac{1}{13}\)
(C) \(\dfrac{1}{13}\)
(D) \(\dfrac{1}{4}\)
▶️ Answer/Explanation
Detailed solution
For an inverse \(g=f^{-1}\), \(g'(a)=\dfrac{1}{f'(g(a))}\). Since \((2,1)\) lies on \(g\), we have \(g(2)=1\) and \(f(1)=2\).
Compute \(f'(x)=3x^{2}+1\Rightarrow f'(1)=3 x 1^{2}+1=4\). Hence \(g'(2)=\dfrac{1}{4}\).
✅ Answer: (D)