AP Calculus AB 3.3 Differentiating Inverse Functions - MCQs - Exam Style Questions
Calc-Ok Question
The table below gives values of the differentiable function \(f\) and its derivative at selected values of \(x\). If \(g\) is the inverse function of \(f\), which of the following is an equation of the line tangent to the graph of \(g\) at the point where \(x=2\) ?
| \(x\) | \(2\) | \(3\) | \(4\) |
| \(f(x)\) | \(1\) | \(2\) | \(6\) |
| \(f'(x)\) | \(4\) | \(5\) | \(3\) |
(A) \(y=-\tfrac{1}{5}(x-2)+3\)
(B) \(y=-\tfrac{1}{4}(x-2)+1\)
(C) \(y=\tfrac{1}{5}(x-2)+3\)
(D) \(y=4(x-2)+1\)
▶️ Answer/Explanation
Inverse-function tangent
Since \(f(3)=2\), the inverse satisfies \(g(2)=3\) so the point is \((2,3)\).
Slope of inverse: \(g'(x)=\dfrac{1}{f'(g(x))}\).
Thus \(g'(2)=\dfrac{1}{f'(3)}=\dfrac{1}{5}\).
Tangent line at \((2,3)\): \(y-3=\dfrac{1}{5}(x-2)\).
So \(y=\dfrac{1}{5}(x-2)+3\).
✅ Answer: (C)