Question 1

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

Accepted Answer

A)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$ 
B)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
C)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$

Question 2

You are given function $f$ defined on an interval $[ a , b ]$, the number $n$ of subintervals of equal length $\Delta x = \frac { ( b - a ) } { n }$, and the evaluation points $c _ { k }$ in $\left[ x _ { k - 1 } , x _ { k } \right]$.
(a) Sketch the graph of $f$ and the rectangles associated with the Riemann sum for $f$ on $[ a , b ]$, and
(b) find the Riemann sum.
$f ( x ) = 2 x - 3 , \quad [ 0,2 ] , \quad n = 4 , \quad c _ { k }$ is the right endpoint

Accepted Answer

The answer of You are given function $f$ defined on...

Question 3

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

Accepted Answer

A)  $1 - 10 \ln x + C$ 
B) 
$x - 10 \ln x - \frac { 25 } { x } + C$ 
C)  $\ln x - 25 x + C$ 
D) 
$x - \frac { 1 } { 10 \ln x } + C$ 
E)  none of these

Question 4

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

Accepted Answer

A)  $\frac { 5 \sqrt { x } } { x ^ { 2 } + 1 }$ 
B)  $\frac { \sqrt { x ^ { 2 } + 1 } } { 5 }$ 
C) 
$$\frac { 5 \sqrt { x } } { 2 \left( x ^ { 2 } + 1 \right) }$$ 
D)  $\frac { \sqrt { x + 5 } } { x ^ { 2 } + 5 }$ 
E)  none of these

Question 5

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

Accepted Answer

A)  $\frac { \sqrt { x + 1 } } { x ^ { 2 } + 2 }$ 
B)  $\frac { \sqrt { x ^ { 2 } + 1 } } { 2 }$ 
C) 
$\frac { \sqrt { x } } { 2 \left( x ^ { 2 } + 1 \right) }$ 
D)  $\frac { \sqrt { x } } { x ^ { 2 } + 1 }$ 
E)  none of these

Question 6

$$\text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. }$$

Accepted Answer

The answer of \[\text { Evaluate } \int _ {...

Question 7

Evaluate the integral $\int _ { 0 } ^ { z / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$.

Accepted Answer

A)  $5 ( e - 1 )$ 
B)  $20 \left( e ^ { 5 } - 1 \right)$ 
C)  $\frac { e - 1 } { 5 }$ 
D)  $\frac { e ^ { 5 } - 1 } { 20 }$

Question 8

A table of values of an increasing function $f ( x )$ is shown. Use the table to find an upper estimate of:
$\begin{array}{|c|c|}
\hline x & f(x) \
\hline 0 & -46 \
\hline 5 & -30 \
\hline 10 & -20 \
\hline 15 & 4 \
\hline 20 & 15 \
\hline 25 & 20 \
\hline
\end{array}$

Accepted Answer

The answer of A table of values of an increasing...

Question 9

Evaluate the definite integral.
$$\int _ { 0 } ^ { x / 6 } \sin 8 t d t$$

Accepted Answer

A)  $0.188$ 
B)  $3.25$ 
C)  $- 0.045$ 
D)  $0.35$ 
E)  $2.891$

Question 10

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 $n = 10$.

Accepted Answer

The answer of The velocity graph of a car accelerating...

Question 11

Find the indefinite integral. $$\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x$$

Accepted Answer

A)  $\cos ^ { 2 } x + C$ 
B)  $\tan x + C$ 
C)  $\csc ^ { 2 } x + C$ 
D)  $- \cot x + C$

Question 12

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

Accepted Answer

The answer of Evaluate the indefinite integral. \[\int \sec ^...

Question 13

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

Accepted Answer

A)  $- e ^ { \cos x } \sin x + C$ 
B)  $- 7 e ^ { \cos x } + C$ 
C)  $e ^ { 7 \sin x } + C$ 
D)  $7 e ^ { \cos x } \sin x + C$ 
E)  $- 7 \sin \left( e ^ { \cos x } \right) + C$

Question 14

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, $F ^ { t } = f$.

Accepted Answer

The answer of Let \(F ( x ) = \int...

Question 15

If $\int _ { 0 } ^ { 6 } f ( x ) d x = 13$ and $\int _ { 0 } ^ { 4 } f ( x ) d x = 7$, find $\int _ { 4 } ^ { 6 } f ( x ) d x$

Accepted Answer

A)  7 
B) -13 
C)  20 
D)  6 
E)  13

Question 16

Evaluate the integral.
$\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t$

Accepted Answer

A)  $\frac { 4 \pi } { 5 }$ 
B)  $\frac { \pi } { 10 }$ 
C)  $\frac { \pi } { 24 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $10 \pi$

Question 17

Express the integral as a limit of sums.Then evaluate the limit. $$\int _ { 0 } ^ { x } \sin 3 x d x$$

Accepted Answer

The answer of Express the integral as a limit of...

Question 18

The acceleration function of a body moving along a coordinate line is $a ( t ) = - 7 \cos 2 t - 8 \sin 2 t \quad t \geq 0$ Find its velocity and position functions at any time $t$ if it is located at the origin and has an initial velocity of $4 \mathrm {~m} / \mathrm { sec }$.

Accepted Answer

The answer of The acceleration function of a body moving...

Question 19

Express the limit as a definite integral on the given interval. $$\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , [ 6,12 ]$$

Accepted Answer

The answer of Express the limit as a definite integral...

Question 20

The marginal cost of manufacturing x yards of a certain fabric is $$C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$$
in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

Accepted Answer

A)  $\$ 188,000.00$ 
B)  $\$ 198,000.00$ 
C)  $\$ 178,000.00$ 
D)  $\$ 218,000.00$ 
E)  $\$ 208,000.00$

Exam 4: Applications of Differentiation

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

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

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

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

$\text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. }$

Evaluate the integral $\int _ { 0 } ^ { z / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$ .

A table of values of an increasing function $f ( x )$ is shown. Use the table to find an upper estimate of: $\begin{array}{|c|c|}\hline x & f(x) \\\hline 0 & -46 \\\hline 5 & -30 \\\hline 10 & -20 \\\hline 15 & 4 \\\hline 20 & 15 \\\hline 25 & 20 \\\hline\end{array}$

Evaluate the definite integral. $\int _ { 0 } ^ { x / 6 } \sin 8 t d t$

Find the indefinite integral. $\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x$

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

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

If $\int _ { 0 } ^ { 6 } f ( x ) d x = 13$ and $\int _ { 0 } ^ { 4 } f ( x ) d x = 7$ , find $\int _ { 4 } ^ { 6 } f ( x ) d x$

Evaluate the integral. $\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t$

Express the integral as a limit of sums.Then evaluate the limit. $\int _ { 0 } ^ { x } \sin 3 x d x$

The acceleration function of a body moving along a coordinate line is $a ( t ) = - 7 \cos 2 t - 8 \sin 2 t \quad t \geq 0$ Find its velocity and position functions at any time $t$ if it is located at the origin and has an initial velocity of $4 \mathrm {~m} / \mathrm { sec }$ .

Express the limit as a definite integral on the given interval. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , [ 6,12 ]$

The marginal cost of manufacturing x yards of a certain fabric is $C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$ in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

Exam 4: Applications of Differentiation

Find the integral using an appropriate trigonometric substitution. ∫x3x2+36dx\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x∫x2+36​x3​dx

Evaluate the integral if it exists. ∫(5−xx) 2dx\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x∫(x5−x​) 2dx

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x) =∫1x5z2z4+1dzh ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { 5 z ^ { 2 } } { z ^ { 4 } + 1 } d zh(x) =∫1x​​z4+15z2​dz

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x) =∫1xz2z4+1dzh ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d zh(x) =∫1x​​z4+1z2​dz

Evaluate ∫03(x+9−x2)dx by interpreting it in terms of areas. \text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. } Evaluate ∫03​(x+9−x2​)dx by interpreting it in terms of areas.

Evaluate the integral ∫0z/20etan⁡5xcos⁡25xdx\int _ { 0 } ^ { z / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x∫0z/20​cos25xetan5x​dx .

Evaluate the definite integral. ∫0x/6sin⁡8tdt\int _ { 0 } ^ { x / 6 } \sin 8 t d t∫0x/6​sin8tdt

The velocity graph of a car accelerating from rest to a speed of 7 km/h7 \mathrm {~km} / \mathrm { h }7 km/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=10n = 10n=10 .

Find the indefinite integral. ∫cot⁡2xcos⁡2xdx\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x∫cos2xcot2x​dx

Evaluate the indefinite integral. ∫sec⁡2xtan⁡xdx\int \sec ^ { 2 } x \tan x d x∫sec2xtanxdx

Evaluate the indefinite integral. ∫7ecos⁡xsin⁡xdx\int 7 e ^ { \cos x } \sin x d x∫7ecosxsinxdx

If ∫06f(x) dx=13\int _ { 0 } ^ { 6 } f ( x ) d x = 13∫06​f(x) dx=13 and ∫04f(x) dx=7\int _ { 0 } ^ { 4 } f ( x ) d x = 7∫04​f(x) dx=7 , find ∫46f(x) dx\int _ { 4 } ^ { 6 } f ( x ) d x∫46​f(x) dx

Evaluate the integral. ∫06136+t2dt\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t∫06​36+t21​dt

Express the integral as a limit of sums.Then evaluate the limit. ∫0xsin⁡3xdx\int _ { 0 } ^ { x } \sin 3 x d x∫0x​sin3xdx

Express the limit as a definite integral on the given interval. lim⁡n→∞∑i=1n(6risin⁡ri)Δr,[6,12]\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , [ 6,12 ]limn→∞​∑i=1n​(6ri​sinri​)Δr,[6,12]

The marginal cost of manufacturing x yards of a certain fabric is Ct(x) =3−0.01x+0.000006x2C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }Ct(x) =3−0.01x+0.000006x2 in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

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

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

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

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

$\text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. }$

Evaluate the integral $\int _ { 0 } ^ { z / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$ .

Evaluate the definite integral. $\int _ { 0 } ^ { x / 6 } \sin 8 t d t$

Find the indefinite integral. $\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x$

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

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

If $\int _ { 0 } ^ { 6 } f ( x ) d x = 13$ and $\int _ { 0 } ^ { 4 } f ( x ) d x = 7$ , find $\int _ { 4 } ^ { 6 } f ( x ) d x$

Evaluate the integral. $\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t$

Express the integral as a limit of sums.Then evaluate the limit. $\int _ { 0 } ^ { x } \sin 3 x d x$

Express the limit as a definite integral on the given interval. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , [ 6,12 ]$

The marginal cost of manufacturing x yards of a certain fabric is $C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$ in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.