Question 1

Find the local and absolute extreme values of the function on the given interval. $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 4$ , $[ 2,4 ]$

Accepted Answer

Max $( 4,8 )$ , Min $( 3,4 )$

Question 2

Determine whether the series is convergent or divergent by expressing $s _ { n }$ as a telescoping sum. If it is convergent, find its sum. $\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { 2 } + 2 n }$ .

Accepted Answer

A)  $\frac { 15 } { 2 }$ 
B)  diverges 
C)  $2$ 
D)  $\frac { 1 } { 10 }$ 
E)  $\frac { 2 } { 15 }$ 
A)  $\frac { 15 } { 2 }$ 
B)  diverges 
C)  $2$ 
D)  $\frac { 1 } { 10 }$ 
E)  $\frac { 2 } { 15 }$ A

Question 3

Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s). $f ( x , y ) = x y z ; \quad x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 30$

Accepted Answer

maximum = $30$ , minimum = $- 30$

Question 4

If an equation of the tangent line to the curve $y = f ( x )$ at the point where $a = 8 \text { is } y = 4 x - 7 \text {, }$ $\text { find } f ( 8 ) \text { and } f ^ { \prime } ( 8 ) \text {. }$

Accepted Answer

A)  $\begin{array} { l } 
f ( 2 ) = 25 \
f ^ { \prime } ( 2 ) = 3
\end{array}$ 
B)  $\begin{array} { l } 
f ( 2 ) = - 8 \
f ^ { \prime } ( 2 ) = 8
\end{array}$ 
C)  $\begin{array} { l } 
f ( 2 ) = 7 \
f ^ { \prime } ( 2 ) = 4
\end{array}$ 
D)  $\begin{array} { l } 
f ( 2 ) = 25 \
f ^ { \prime } ( 2 ) = 4
\end{array}$ 
E)  None of these 
A)  $\begin{array} { l } 
f ( 2 ) = 25 \
f ^ { \prime } ( 2 ) = 3
\end{array}$ 
B)  $\begin{array} { l } 
f ( 2 ) = - 8 \
f ^ { \prime } ( 2 ) = 8
\end{array}$ 
C)  $\begin{array} { l } 
f ( 2 ) = 7 \
f ^ { \prime } ( 2 ) = 4
\end{array}$ 
D)  $\begin{array} { l } 
f ( 2 ) = 25 \
f ^ { \prime } ( 2 ) = 4
\end{array}$ 
E)  None of these

Question 5

The graphs of $f ( x )$ and $g ( x )$ are given. For what values of x is $f ( x ) = g ( x )$ ?

Accepted Answer

A)  $- 1$ 
B)  $- 2,5$ 
C)  -4, 12 
D)  0 
E)  4, 2 
A)  $- 1$ 
B)  $- 2,5$ 
C)  -4, 12 
D)  0 
E)  4, 2

Question 6

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius $6$ .

Accepted Answer

A)  $\frac { 4374 } { 7 } \pi$ 
B)  $\frac { 559872 } { 7 } \pi$ 
C)  $\frac { 1119744 } { 7 } \pi$ 
D)  $\frac { 4374 } { 7 }$ 
E)  None of these 
A)  $\frac { 4374 } { 7 } \pi$ 
B)  $\frac { 559872 } { 7 } \pi$ 
C)  $\frac { 1119744 } { 7 } \pi$ 
D)  $\frac { 4374 } { 7 }$ 
E)  None of these

Question 7

If $\mathbf { r } ( t ) = 5 \mathbf { i } + 3 t \cos \pi \mathbf { j } + 4 \sin \pi t \mathbf { k }$ , evaluate $\int _ { 0 } ^ { 1 } r ( t ) d t$ .

Accepted Answer

A)  $5 \mathbf { i } - \frac { 6 } { \pi ^ { 2 } } \mathbf { j } - \frac { 1 } { \pi } \mathbf { k }$ 
B)  $\mathbf { i } - \frac { 6 } { \pi ^ { 2 } } \mathbf { j } + \frac { 1 } { \pi } \mathbf { k }$ 
C)  $\mathbf { i } + \frac { 6 } { \pi ^ { 2 } } \mathbf { j } - \frac { 8 } { \pi } \mathbf { k }$ 
D)  $5 \mathbf { i } - \frac { 6 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$ 
E)  $5 \mathbf { i } + \frac { 6 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$ 
A)  $5 \mathbf { i } - \frac { 6 } { \pi ^ { 2 } } \mathbf { j } - \frac { 1 } { \pi } \mathbf { k }$ 
B)  $\mathbf { i } - \frac { 6 } { \pi ^ { 2 } } \mathbf { j } + \frac { 1 } { \pi } \mathbf { k }$ 
C)  $\mathbf { i } + \frac { 6 } { \pi ^ { 2 } } \mathbf { j } - \frac { 8 } { \pi } \mathbf { k }$ 
D)  $5 \mathbf { i } - \frac { 6 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$ 
E)  $5 \mathbf { i } + \frac { 6 } { \pi ^ { 2 } } \mathbf { j } + \frac { 8 } { \pi } \mathbf { k }$

Question 8

The graphs of $f ( x )$ and $g ( x )$ are given.
Find the values of $f ( - 4 )$ and $g ( 3 )$ .

Accepted Answer

The answer of The graphs of $f ( x )$...

Question 9

The acceleration function (in m / $s ^ { 2 }$ ) and the initial velocity are given for a particle moving along a line. Find the velocity at time t and the distance traveled during the given time interval. $a ( t ) = t + 4 , v ( 0 ) = 3,0 \leq t \leq 10$

Accepted Answer

A)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 3 t \mathrm {~m} / \mathrm { s } , 571 \frac { 2 } { 3 } \mathrm {~m}$ 
B)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 3 \mathrm {~m} / \mathrm { s } , 596 \frac { 2 } { 3 } \mathrm {~m}$ 
C)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 4 t + 3 \mathrm {~m} / \mathrm { s } , 496 \frac { 2 } { 3 } \mathrm {~m}$ 
D)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 3 t \mathrm {~m} / \mathrm { s } , 546 \frac { 2 } { 3 } \mathrm {~m}$ 
E)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 4 t + 3 \mathrm {~m} / \mathrm { s } , 396 \frac { 2 } { 3 } \mathrm {~m}$ 
A)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 3 t \mathrm {~m} / \mathrm { s } , 571 \frac { 2 } { 3 } \mathrm {~m}$ 
B)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 3 \mathrm {~m} / \mathrm { s } , 596 \frac { 2 } { 3 } \mathrm {~m}$ 
C)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 4 t + 3 \mathrm {~m} / \mathrm { s } , 496 \frac { 2 } { 3 } \mathrm {~m}$ 
D)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 3 t \mathrm {~m} / \mathrm { s } , 546 \frac { 2 } { 3 } \mathrm {~m}$ 
E)  $v ( t ) = \frac { t ^ { 2 } } { 2 } + 4 t + 3 \mathrm {~m} / \mathrm { s } , 396 \frac { 2 } { 3 } \mathrm {~m}$

Question 10

Find the directional derivative of $f ( x , y ) = 20 \sqrt { x } - y ^ { 3 }$ at the point (1, 3) in the direction toward the point (3, 1).

Accepted Answer

A)  $37 \sqrt { 2 }$ 
B)  $\frac { 37 } { \sqrt { 2 } }$ 
C)  $\sqrt { 2 }$ 
D)  $37$ 
E)  None of these 
A)  $37 \sqrt { 2 }$ 
B)  $\frac { 37 } { \sqrt { 2 } }$ 
C)  $\sqrt { 2 }$ 
D)  $37$ 
E)  None of these

Question 11

Find the volume of the given solid.
Under the paraboloid $z = 5 x ^ { 2 } + 15 y ^ { 2 }$ and above the rectangle $R = [ 0,3 ] \times [ 1,5 ]$ .

Accepted Answer

The answer of Find the volume of the given solid.
Under...

Question 12

Find the curvature of $y = x ^ { 7 }$ .

Accepted Answer

A)  $\frac { | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 1 / 2 } }$ 
B)  $\frac { 42 | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 3 / 2 } }$ 
C)  $\frac { 42 | x | ^ { 5 } } { \left( 1 - 49 x ^ { 12 } \right) ^ { 3 / 2 } }$ 
D)  $\frac { | x | ^ { 5 } } { \left( 1 + x ^ { 12 } \right) ^ { 3 / 2 } }$ 
E)  $\frac { 42 | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 1 / 2 } }$ 
A)  $\frac { | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 1 / 2 } }$ 
B)  $\frac { 42 | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 3 / 2 } }$ 
C)  $\frac { 42 | x | ^ { 5 } } { \left( 1 - 49 x ^ { 12 } \right) ^ { 3 / 2 } }$ 
D)  $\frac { | x | ^ { 5 } } { \left( 1 + x ^ { 12 } \right) ^ { 3 / 2 } }$ 
E)  $\frac { 42 | x | ^ { 5 } } { \left( 1 + 49 x ^ { 12 } \right) ^ { 1 / 2 } }$

Question 13

If $f ( x ) = x ^ { 2 } - x + 4$ , evaluate the difference quotient $\frac { f ( a + h ) - f ( a ) } { h }$ .

Accepted Answer

A)  $2 a - 4$ 
B)  $2 a + h - 4$ 
C)  h 
D)  $2 a - h - 4$ 
E)  None of these 
A)  $2 a - 4$ 
B)  $2 a + h - 4$ 
C)  h 
D)  $2 a - h - 4$ 
E)  None of these

Question 14

Find the sum of the series. $\frac { 2 } { 1.9 } - \frac { 2 ^ { 2 } } { 2.9 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3.9 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4.9 ^ { 4 } } + \ldots$

Accepted Answer

A)  $\frac { 11 e } { 9 }$ 
B)  $\ln \left( \frac { 1 } { 9 } \right)$ 
C)  $\frac { 11 } { 9 }$ 
D)  $\ln \left( \frac { 11 } { 9 } \right)$ 
E)  $\ln \left( \frac { 10 } { 9 } \right)$ 
A)  $\frac { 11 e } { 9 }$ 
B)  $\ln \left( \frac { 1 } { 9 } \right)$ 
C)  $\frac { 11 } { 9 }$ 
D)  $\ln \left( \frac { 11 } { 9 } \right)$ 
E)  $\ln \left( \frac { 10 } { 9 } \right)$

Question 15

Evaluate the integral using the indicated trigonometric substitution. $$\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 36 } } ; x = 6 \sec \theta$$

Accepted Answer

The answer of Evaluate the integral using the indicated trigonometric...

Question 16

Find the volume of the resulting solid if the region under the curve $y = \frac { 1 } { \left( x ^ { 2 } + 3 x + 2 \right) }$ from $x = 0$ to $x = 1$ is rotated about the x-axis. Round your answer to four decimal places.

Accepted Answer

The answer of Find the volume of the resulting solid...

Question 17

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { 6 d x } { x \ln x }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $0$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\text { divergent }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $0$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\text { divergent }$

Question 18

How would you define $f ( 7 )$ in order to make f continuous at $7$ ? $$f ( x ) = \frac { x ^ { 2 } - 4 x - 4 } { x - 3 }$$

Accepted Answer

A)  $f ( 7 ) = - 10$ 
B)  $f ( 7 ) = 0$ 
C)  $f ( 7 ) = 10$ 
D)  $f ( 3 ) = - 6$ 
E)  None of these 
A)  $f ( 7 ) = - 10$ 
B)  $f ( 7 ) = 0$ 
C)  $f ( 7 ) = 10$ 
D)  $f ( 3 ) = - 6$ 
E)  None of these

Question 19

Calculate the given quantities if $\mathbf { a } = \mathbf { i } - 9 \mathbf { j } \text { and } \mathbf { b } = - 9 \mathbf { j } + 4 \mathbf { k }$ I. $\quad \mathbf { a } \cdot \mathbf { b }$
II $\mathbf { a } \times \mathbf { b }$
III. The angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree)

Accepted Answer

\[\begin{array} { l } 
\mathbf

Question 20

The masses $m _ { i }$ are located at the point $P _ { 1 }$ . Find the moments $M _ { x }$ and $M _ { y }$ and the center of mass of the system. $m _ { 1 } = 3 , m _ { 2 } = 7 , m _ { 3 } = 111$ ; $$P _ { 1 } ( 1,5 ) , P _ { 2 } ( 3 , - 2 ) , P _ { 3 } ( - 2 , - 1 )$$

Accepted Answer

A)  $M _ { x } = 18 , M _ { y } = 44 , \left( \frac { 44 } { 21 } , \frac { 18 } { 21 } \right)$ 
B)  $M _ { x } = 44 , M _ { y } = 18 , \left( \frac { 44 } { 21 } , \frac { 18 } { 21 } \right)$ 
C)  $M _ { x } = 18 , M _ { y } = 44 , \left( \frac { 18 } { 21 } , \frac { 44 } { 21 } \right)$ 
D)  $M _ { x } = 44 , M _ { y } = - 18 , \left( - \frac { 44 } { 21 } , \frac { 18 } { 21 } \right)$ 
E)  $M _ { x } = - 44 , M _ { y } = 18 , \left( \frac { 44 } { 21 } , - \frac { 18 } { 21 } \right)$ 
A)  $M _ { x } = 18 , M _ { y } = 44 , \left( \frac { 44 } { 21 } , \frac { 18 } { 21 } \right)$ 
B)  $M _ { x } = 44 , M _ { y } = 18 , \left( \frac { 44 } { 21 } , \frac { 18 } { 21 } \right)$ 
C)  $M _ { x } = 18 , M _ { y } = 44 , \left( \frac { 18 } { 21 } , \frac { 44 } { 21 } \right)$ 
D)  $M _ { x } = 44 , M _ { y } = - 18 , \left( - \frac { 44 } { 21 } , \frac { 18 } { 21 } \right)$ 
E)  $M _ { x } = - 44 , M _ { y } = 18 , \left( \frac { 44 } { 21 } , - \frac { 18 } { 21 } \right)$

Find the local and absolute extreme values of the function on the given interval. $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 4$ , $[ 2,4 ]$

Determine whether the series is convergent or divergent by expressing $s _ { n }$ as a telescoping sum. If it is convergent, find its sum. $\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { 2 } + 2 n }$ .

Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s). $f ( x , y ) = x y z ; \quad x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 30$

If an equation of the tangent line to the curve $y = f ( x )$ at the point where $a = 8 \text { is } y = 4 x - 7 \text {, }$ $\text { find } f ( 8 ) \text { and } f ^ { \prime } ( 8 ) \text {. }$

The graphs of $f ( x )$ and $g ( x )$ are given. For what values of x is $f ( x ) = g ( x )$ ?

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius $6$ .

If $\mathbf { r } ( t ) = 5 \mathbf { i } + 3 t \cos \pi \mathbf { j } + 4 \sin \pi t \mathbf { k }$ , evaluate $\int _ { 0 } ^ { 1 } r ( t ) d t$ .

The graphs of $f ( x )$ and $g ( x )$ are given. Find the values of $f ( - 4 )$ and $g ( 3 )$ .

The acceleration function (in m / $s ^ { 2 }$ ) and the initial velocity are given for a particle moving along a line. Find the velocity at time t and the distance traveled during the given time interval. $a ( t ) = t + 4 , v ( 0 ) = 3,0 \leq t \leq 10$

Find the directional derivative of $f ( x , y ) = 20 \sqrt { x } - y ^ { 3 }$ at the point (1, 3) in the direction toward the point (3, 1).

Find the volume of the given solid. Under the paraboloid $z = 5 x ^ { 2 } + 15 y ^ { 2 }$ and above the rectangle $R = [ 0,3 ] \times [ 1,5 ]$ .

Find the curvature of $y = x ^ { 7 }$ .

If $f ( x ) = x ^ { 2 } - x + 4$ , evaluate the difference quotient $\frac { f ( a + h ) - f ( a ) } { h }$ .

Find the sum of the series. $\frac { 2 } { 1.9 } - \frac { 2 ^ { 2 } } { 2.9 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3.9 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4.9 ^ { 4 } } + \ldots$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 36 } } ; x = 6 \sec \theta$

Find the volume of the resulting solid if the region under the curve $y = \frac { 1 } { \left( x ^ { 2 } + 3 x + 2 \right) }$ from $x = 0$ to $x = 1$ is rotated about the x-axis. Round your answer to four decimal places.

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { 6 d x } { x \ln x }$

How would you define $f ( 7 )$ in order to make f continuous at $7$ ? $f ( x ) = \frac { x ^ { 2 } - 4 x - 4 } { x - 3 }$

The masses $m _ { i }$ are located at the point $P _ { 1 }$ . Find the moments $M _ { x }$ and $M _ { y }$ and the center of mass of the system. $m _ { 1 } = 3 , m _ { 2 } = 7 , m _ { 3 } = 111$ ; $P _ { 1 } ( 1,5 ) , P _ { 2 } ( 3 , - 2 ) , P _ { 3 } ( - 2 , - 1 )$

Functions and Limits

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Exam 18: Final Exam

Find the local and absolute extreme values of the function on the given interval. f(x)=x3−6x2+9x+4f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 4f(x)=x3−6x2+9x+4 , [2,4][ 2,4 ][2,4]

Determine whether the series is convergent or divergent by expressing sns _ { n }sn​ as a telescoping sum. If it is convergent, find its sum. ∑n=1∞10n2+2n\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { 2 } + 2 n }∑n=1∞​n2+2n10​ .

Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s). f(x,y)=xyz;x2+y2+z2=30f ( x , y ) = x y z ; \quad x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 30f(x,y)=xyz;x2+y2+z2=30

If an equation of the tangent line to the curve y=f(x)y = f ( x )y=f(x) at the point where a=8 is y=4x−7, a = 8 \text { is } y = 4 x - 7 \text {, }a=8 is y=4x−7, find f(8) and f′(8). \text { find } f ( 8 ) \text { and } f ^ { \prime } ( 8 ) \text {. } find f(8) and f′(8).

The graphs of f(x)f ( x )f(x) and g(x)g ( x )g(x) are given. For what values of x is f(x)=g(x)f ( x ) = g ( x )f(x)=g(x) ?

Use spherical coordinates. Evaluate ∭B(x2+y2+z2)2dV\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V∭B​(x2+y2+z2)2dV , where BBB is the ball with center the origin and radius 666 .

If r(t)=5i+3tcos⁡πj+4sin⁡πtk\mathbf { r } ( t ) = 5 \mathbf { i } + 3 t \cos \pi \mathbf { j } + 4 \sin \pi t \mathbf { k }r(t)=5i+3tcosπj+4sinπtk , evaluate ∫01r(t)dt\int _ { 0 } ^ { 1 } r ( t ) d t∫01​r(t)dt .

The graphs of f(x)f ( x )f(x) and g(x)g ( x )g(x) are given. Find the values of f(−4)f ( - 4 )f(−4) and g(3)g ( 3 )g(3) .

Find the directional derivative of f(x,y)=20x−y3f ( x , y ) = 20 \sqrt { x } - y ^ { 3 }f(x,y)=20x​−y3 at the point (1, 3) in the direction toward the point (3, 1).

Find the volume of the given solid. Under the paraboloid z=5x2+15y2z = 5 x ^ { 2 } + 15 y ^ { 2 }z=5x2+15y2 and above the rectangle R=[0,3]×[1,5]R = [ 0,3 ] \times [ 1,5 ]R=[0,3]×[1,5] .

Find the curvature of y=x7y = x ^ { 7 }y=x7 .

If f(x)=x2−x+4f ( x ) = x ^ { 2 } - x + 4f(x)=x2−x+4 , evaluate the difference quotient f(a+h)−f(a)h\frac { f ( a + h ) - f ( a ) } { h }hf(a+h)−f(a)​ .

Find the sum of the series. 21.9−222.92+233.93−244.94+…\frac { 2 } { 1.9 } - \frac { 2 ^ { 2 } } { 2.9 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3.9 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4.9 ^ { 4 } } + \ldots1.92​−2.9222​+3.9323​−4.9424​+…

Evaluate the integral using the indicated trigonometric substitution. ∫dxx2x2−36;x=6sec⁡θ\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 36 } } ; x = 6 \sec \theta∫x2x2−36​dx​;x=6secθ

Find the volume of the resulting solid if the region under the curve y=1(x2+3x+2)y = \frac { 1 } { \left( x ^ { 2 } + 3 x + 2 \right) }y=(x2+3x+2)1​ from x=0x = 0x=0 to x=1x = 1x=1 is rotated about the x-axis. Round your answer to four decimal places.

Evaluate the integral. ∫1∞6dxxln⁡x\int _ { 1 } ^ { \infty } \frac { 6 d x } { x \ln x }∫1∞​xlnx6dx​

How would you define f(7)f ( 7 )f(7) in order to make f continuous at 777 ? f(x)=x2−4x−4x−3f ( x ) = \frac { x ^ { 2 } - 4 x - 4 } { x - 3 }f(x)=x−3x2−4x−4​

Functions and Limits

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Techniques of Integration

Further Applications of Integration

Differential Equations

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Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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Find the local and absolute extreme values of the function on the given interval. $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 4$ , $[ 2,4 ]$

Determine whether the series is convergent or divergent by expressing $s _ { n }$ as a telescoping sum. If it is convergent, find its sum. $\sum _ { n = 1 } ^ { \infty } \frac { 10 } { n ^ { 2 } + 2 n }$ .

Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s). $f ( x , y ) = x y z ; \quad x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 30$

If an equation of the tangent line to the curve $y = f ( x )$ at the point where $a = 8 \text { is } y = 4 x - 7 \text {, }$ $\text { find } f ( 8 ) \text { and } f ^ { \prime } ( 8 ) \text {. }$

The graphs of $f ( x )$ and $g ( x )$ are given. For what values of x is $f ( x ) = g ( x )$ ?

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius $6$ .

If $\mathbf { r } ( t ) = 5 \mathbf { i } + 3 t \cos \pi \mathbf { j } + 4 \sin \pi t \mathbf { k }$ , evaluate $\int _ { 0 } ^ { 1 } r ( t ) d t$ .

The graphs of $f ( x )$ and $g ( x )$ are given. Find the values of $f ( - 4 )$ and $g ( 3 )$ .

Find the directional derivative of $f ( x , y ) = 20 \sqrt { x } - y ^ { 3 }$ at the point (1, 3) in the direction toward the point (3, 1).

Find the volume of the given solid. Under the paraboloid $z = 5 x ^ { 2 } + 15 y ^ { 2 }$ and above the rectangle $R = [ 0,3 ] \times [ 1,5 ]$ .

Find the curvature of $y = x ^ { 7 }$ .

If $f ( x ) = x ^ { 2 } - x + 4$ , evaluate the difference quotient $\frac { f ( a + h ) - f ( a ) } { h }$ .

Find the sum of the series. $\frac { 2 } { 1.9 } - \frac { 2 ^ { 2 } } { 2.9 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3.9 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4.9 ^ { 4 } } + \ldots$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 36 } } ; x = 6 \sec \theta$

Find the volume of the resulting solid if the region under the curve $y = \frac { 1 } { \left( x ^ { 2 } + 3 x + 2 \right) }$ from $x = 0$ to $x = 1$ is rotated about the x-axis. Round your answer to four decimal places.

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { 6 d x } { x \ln x }$

How would you define $f ( 7 )$ in order to make f continuous at $7$ ? $f ( x ) = \frac { x ^ { 2 } - 4 x - 4 } { x - 3 }$