AP Calculus BC Unit 3: Differentiation: Composite, Implicit, and Inverse Functions Practice Exam

1. Let y = $\sqrt{1 + \cos(3x)} $. Which of the following is an expression for $\frac{dy}{dx} $ ?

2. The line y = 4x - 5 is tangent to the graph of the function f at x = 3 . If g is a differentiable function defined by g(x) = f(x^2 - 1) , what is the value of g'(2) ?

3. Consider the curve given by the equation $\sin(x+y) = 3x - 2y$. Which of the following expressions represents $\frac{dy}{dx}$?

4. For the curve defined by the equation $xy^2 - x^3 = 10$, what is the slope of the tangent line at the point $(2, 3)$?

5. At which points does the curve given by $x^2 + xy + y^2 = 27$ have a horizontal tangent line?

6. The table below gives selected values for a differentiable and decreasing function $f$ and its derivative $f'$. $$ \begin{array}{|c|c|c|} \hline x & f(x) & f'(x) \\ \hline -1 & 4 & -2 \\ \hline 0 & 3 & -3 \\ \hline 3 & -1 & -5 \\ \hline \end{array} $$ If $g(x) = f^{-1}(x)$ for all $x$, what is the value of $g'(3)$?

7. Let $f$ be the function defined by $f(x) = x^3 + 2x - 1$. If $h$ is the inverse function of $f$, what is the value of $h'(2)$?

8. The graph of the function $f$ is shown in the figure above. The line tangent to the graph of $f$ at the point $(4, 6)$ passes through the point $(7, 0)$. If $g$ is the inverse function of $f$, what is the value of $g'(6)$?

9. If $y = \arcsin(5x)$, then $\frac{dy}{dx} =$

10. Let $f$ be a differentiable function such that $f(1) = \frac{1}{2}$ and $f'(1) = 2$. If $g(x) = \arccos(f(x))$, what is the value of $g'(1)$?

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

12. The functions $f$ and $g$ are differentiable for all real numbers, and $g(x) = f^{-1}(x)$. The table below gives values of $f$ and its derivative $f'$ at selected values of $x$. | $x$ | $f(x)$ | $f'(x)$ | |---|---|---| | 1 | 3 | -2 | | 3 | 4 | 5 | If $h(x) = g(2x+1)$, what is the value of $h'(1)$?

13. Let $h(x) = f(g(x))$, where $f$ and $g$ are twice-differentiable functions. The table below gives values for the functions and their derivatives at selected values of $x$. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline x & f(x) & f'(x) & f''(x) & g(x) & g'(x) & g''(x) \\ \hline 2 & 1 & 3 & -1 & 3 & 2 & 1 \\ \hline 3 & 4 & 5 & -2 & 0 & -1 & 0 \\ \hline \end{array} $$ What is the value of $h''(2)$?

14. Consider the curve defined by the equation $x^2 + y^2 = 25$. What is the value of $\frac{d^2y}{dx^2}$ at the point $(3, 4)$?