Implicit and inverse-function differentiation Knowledge-Based Practice

Questions

1. 1.

   A curve is defined implicitly by the equation $x e^y + y = 1$. What is the slope of the normal line to the curve at the point $(1, 0)$?

   • A) $-\frac{1}{2}$
   • B) 2
   • C) -2
   • D) $\frac{1}{2}$
2. 2.

   A curve is defined implicitly by the equation $y^2 = x^3 - 3x$. At which of the following points does the curve have a horizontal tangent line?

   • A) $(0, 0)$
   • B) $(-1, \sqrt{2})$
   • C) $(\sqrt{3}, 0)$
   • D) $(1, \sqrt{2})$
3. 3.

   A curve is defined implicitly by the equation $x^2 - xy + y^2 = 3$. What is the value of $\frac{d^2y}{dx^2}$ at the point $(1, 2)$?

   • A) $-\frac{2}{3}$
   • B) $\frac{2}{3}$
   • C) $-\frac{1}{3}$
   • D) $0$
4. 4.

   A curve is defined implicitly by the equation $x^2 y + 2y^3 = 6$. What is the value of $\frac{dy}{dx}$ at the point $(2, 1)$?

   • A) $\frac{2}{5}$
   • B) $-\frac{2}{5}$
   • C) $-\frac{4}{5}$
   • D) $-\frac{1}{2}$
5. 5.

   Let $f(x) = x^3 + 2x - 1$ be an invertible, differentiable function. What is the value of $(f^{-1})'(2)$?

   • A) 5
   • B) $\frac{1}{14}$
   • C) $\frac{1}{3}$
   • D) $\frac{1}{5}$

Answers and step-by-step solutions appear after you finish the test.