3. The graph of the function $f$ shown above consists of six line segments. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
(a) Find $g ( 4 ) , g ^ { \prime } ( 4 )$, and $g ^ { \prime \prime } ( 4 )$.
(b) Does $g$ have a relative minimum, a relative maximum, or neither at $x = 1$ ? Justify your answer.
(c) Suppose that $f$ is defined for all real numbers $x$ and is periodic with a period of length 5 . The graph above shows two periods of $f$. Given that $g ( 5 ) = 2$, find $g ( 10 )$ and write an equation for the line tangent to the graph of $g$ at $x = 108$.
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
END OF PART A OF SECTION II
No calculator is allowed for these problems.
| \begin{tabular}{ c } $t$ |
| (seconds) |
& 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \hline
| $v ( t )$ |
| (feet per second) |
& 5 & 14 & 22 & 29 & 35 & 40 & 44 & 47 & 49 \hline \end{tabular}
- Rocket $A$ has positive velocity $v ( t )$ after being launched upward from an initial height of 0 feet at time $t = 0$ seconds. The velocity of the rocket is recorded for selected values of $t$ over the interval $0 \leq t \leq 80$ seconds, as shown in the table above.
(a) Find the average acceleration of rocket $A$ over the time interval $0 \leq t \leq 80$ seconds. Indicate units of measure.
(b) Using correct units, explain the meaning of $\int _ { 10 } ^ { 70 } v ( t ) d t$ in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate $\int _ { 10 } ^ { 70 } v ( t ) d t$.
(c) Rocket $B$ is launched upward with an acceleration of $a ( t ) = \frac { 3 } { \sqrt { t + 1 } }$ feet per second per second. At time $t = 0$ seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time $t = 80$ seconds? Explain your answer.
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
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