Exam 4: Applications of Differentiation

Evaluate the indefinite integral. $\int \cos ^ { 9 } x \sin x d x$

Evaluate the integral $\int _ { 1 } ^ { 9 } \frac { 8 x ^ { 2 } - 3 x + 7 } { x } d x$ .

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

Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after $t$ weeks is $\frac { d x } { d t } = 5700 \left( 1 - \frac { 160 } { ( t + 22 ) ^ { 2 } } \right)$ calculators per week. Production approaches 5,700 per week as time goes on, but the initial production is lower because of the workers' unfamiliarity with the new techniques. Find the number of calculators produced from the beginning of the third week to the end of the fourth week. Round the answer to the nearest integer.

Approximate the area under the curve $y = \frac { 5 } { x ^ { 2 } }$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Let $F ( x ) = \int _ { 5 } ^ { x } 5 t ^ { 2 } d t$ . a. Use Part 1 of the Fundamental Theorem of Calculus to find $F ^ { \prime } ( x )$ . b. Use Part 2 of the Fundamental Theorem of Calculus to integrate $\int _ { 5 } ^ { x } 5 t ^ { 2 } d t$ to obtain an alternative expression for $F ( x )$ . c. Differentiate the expression for $F ( x )$ found in part (b). The Fundamental Theorem of Calculus, Part 1 If $f$ is continuous on $[ a , b ]$ , then the function $F$ defined by $F ( x ) = \int _ { a } ^ { x } f ( t ) d t \quad a \leq x \leq b$ is differentiable on $( a , b )$ , and $F ^ { \prime } ( x ) = \frac { d } { d x } \int _ { a } ^ { x } f ( t ) d t = f ( x )$ The Fundamental Theorem of Calculus, Part 2 If $f$ is continuous on $[ a , b ]$ , then $\int _ { a } ^ { b } f ( x ) d x = F ( b ) - F ( a )$ where $F$ is any antiderivative of $f$ , that is, $F ^ { \prime } = f$ .

Evaluate the integral by making the given substitution. $\int x ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x , \quad u = x ^ { 3 } + 2$

Approximate the area under the curve $y = \sin x$ from 0 to $\frac { \pi } { 2 }$ using 8 approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt { 9 - x ^ { 2 } } d x$

$\text { Use the graph of } f \text { shown in the figure to evaluate the integral by interpreting it geometrically. }$ $\int_{-4}^{7} f(x) d x$ Select the correct answer.

The velocity of a car was read from its speedometer at ten-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. t() v(/) t() v(/) 0 0 60 59 10 32 70 62 20 49 80 71 30 32 90 44 40 44 100 45 50 42

Find the area of the region that lies beneath the given curve. $y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }$

Find the integral using the indicated substitution. $\int ( 4 x + 3 ) ^ { 6 } d x , \quad u = 4 x + 3$ Select the correct answer.

Evaluate the indefinite integral. $\int \frac { e ^ { x } } { e ^ { x } + 5 } d x$

Evaluate the integral. $\int _ { 1 } ^ { 36 } \frac { 1 } { \sqrt { x } } d x$

Evaluate the Riemann sum for $f ( r ) = 6 - r ^ { 2 } , 0 \leq r \leq 2$ with four subintervals, taking the sample points to be right endpoints.

Evaluate the indefinite integral. $\int \sec ^ { 2 } x \tan x d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$ Select the correct answer.

Find the derivative of the function. $F ( x ) = \int _ { x } ^ { x ^ { 2 } } e ^ { t ^ { 5 } } d t$