ap-calculus-ab 2011 Q6

ap-calculus-ab · Usa · free-response Differentiating Transcendental Functions Piecewise function analysis with transcendental components

Let $f$ be a function defined by $$f(x) = \begin{cases} 1 - 2\sin x & \text{for } x \leq 0 \\ e^{-4x} & \text{for } x > 0. \end{cases}$$
(a) Show that $f$ is continuous at $x = 0$.
(b) For $x \neq 0$, express $f'(x)$ as a piecewise-defined function. Find the value of $x$ for which $f'(x) = -3$.
(c) Find the average value of $f$ on the interval $[-1, 1]$.

Let $f$ be a function defined by
$$f(x) = \begin{cases} 1 - 2\sin x & \text{for } x \leq 0 \\ e^{-4x} & \text{for } x > 0. \end{cases}$$

(a) Show that $f$ is continuous at $x = 0$.

(b) For $x \neq 0$, express $f'(x)$ as a piecewise-defined function. Find the value of $x$ for which $f'(x) = -3$.

(c) Find the average value of $f$ on the interval $[-1, 1]$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6