Implicit and inverse-function differentiation Knowledge-Based Download PDF

Questions

1. 1.

   What is the equation of the normal line to the curve $x^3 + y^3 - 9xy = 0$ at the point $(2, 4)$?

   • A) $5x + 4y = 26$
   • B) $4x - 5y = -12$
   • C) $5x - 4y = -6$
   • D) $4x + 5y = 28$
2. 2.

   At which point on the curve $x^2 + 4y^2 = 16$ in the first quadrant is the tangent line parallel to the line $x + 2y = 0$?

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

   For the curve defined by the equation $x^3 + y^3 = 2$, what is the value of $\frac{d^2y}{dx^2}$ at the point $(1, 1)$?

   • A) $-2$
   • B) $-4$
   • C) $2$
   • D) $4$
4. 4.

   Let $f(x) = x^3 + 3x - 2$ be a one-to-one differentiable function, and let $g(x) = f^{-1}(x)$ denote its inverse. What is the value of $g'(2)$?

   • A) $\frac{1}{15}$
   • B) $\frac{1}{6}$
   • C) $6$
   • D) $\frac{1}{3}$
5. 5.

   Consider the curve defined by the equation $x^2 y + 2y^3 = 6$. What is the slope of the tangent line to the curve at the point $(2, 1)$?

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