Chapter 5, Problem 56RE

Area functions and the Fundamental Theorem Consider the function

$f(t)=\{\begin{array}{ll}t\hfill & if-2\le t<0\hfill \\ \frac{t^2}{2}\hfill & if0\le t\le 2\hfill \end{array}$

and its graph shown below.

Let $F(x)={\displaystyle {\int }_{-1}^{x}f(t)dt}$ and ${\displaystyle {\int }_{-2}^{x}f(t)dt}$.

56.

1. a. Evaluate G(−1) and G(1).
2. b. Use the Fundamental Theorem to find an expression for G′(x), for −2 ≤ x ≤ 0.
3. c. Use the Fundamental Theorem to find an expression for G′(x), for 0 ≤ x ≤ 2.
4. d. Evaluate G′(0) and G′(1). Interpret these values.
5. e. Find a constant C such that F(x) = G(x) + C.