Exam 4: Applications of Differentiation

Find $g ^ { \prime } ( x )$ by evaluating the integral using Part 2 of the Fundamental Theorem and then differentiating. $g ( x ) = \int _ { x } ^ { x } ( 6 + \cos t ) d t$

Find the integral. $\int \sin ^ { 3 } x \cos ^ { 6 } x d x$

The velocity graph of a car accelerating from rest to a speed of $7 \mathrm {~km} / \mathrm { h }$ over a period of 10 seconds is shown. Estimate to the nearest integer the distance traveled during this period. Use a right sum with $n = 10$ .

If $h ^ { \prime }$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 3 and 7 ?

$\text { Find the indefinite integral } \int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x \text {. }$ Select the correct answer.

Evaluate the integral. $\int _ { \pi / 6 } ^ { \pi / 2 } \sin t d t$ Select the correct answer.

Find the integral. $\int \cos ^ { 5 } x \sin ^ { 4 } x d x$

Find the integral using the indicated substitution. $\int \tan ^ { 2 } x \sec ^ { 2 } x d x , \quad u = \tan x$ Select the correct answer.

Find the area of the region that lies under the given curve. Round the answer to three decimal places. $y = \sqrt { 5 x + 2 } , 0 \leq x \leq 1$ Select the correct answer.

Find the indefinite integral. $\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { 5 z ^ { 2 } } { z ^ { 4 } + 1 } d z$ Select the correct answer.

Evaluate $\frac { d } { d x } \int _ { 0 } ^ { x } \left( e ^ { \arcsin t } \right) d t$

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]$ Select the correct answer.

Evaluate the indefinite integral. $\int 7 e ^ { \cos x } \sin x d x$

Evaluate the integral if it exists. $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$

Use the Midpoint Rule with $n = 6$ to approximate the integral. Round the answer to 3 decimal places. $\int _ { 0 } ^ { 12 } 2 \sin \sqrt { t } d t$

Evaluate the integral by making the given substitution $\int \cos 3 x d x , u = 3 x$

Find the limit. $\lim _ { h \rightarrow 0 } \frac { 1 } { h } \int _ { 2 } ^ { 2 + h } \sqrt { 1 + t ^ { 3 } } d t$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { e ^ { 2 x } } { 1 + e ^ { 4 x } } d x$