Question 1

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. $f ( x ) = 2 \sqrt { x }$ , $[ 0,9 ]$

Accepted Answer

A)  $c =$ $\frac { 9 } { 4 }$ 
B)  $c = 5$ 
C)  $c = 0$ 
D)  $c = \frac { 1 } { 3 }$ 
E)  None of these 
A)  $c =$ $\frac { 9 } { 4 }$ 
B)  $c = 5$ 
C)  $c = 0$ 
D)  $c = \frac { 1 } { 3 }$ 
E)  None of these

Question 2

Evaluate the integral. $\int x \sin x \cos x d x$

Accepted Answer

A)  $- \frac { 1 } { 2 } \cos x \sin x + \frac { 1 } { 2 } x + C$ 
B)  $- \frac { 1 } { 2 } x \sin ^ { 2 } x + \frac { 1 } { 2 } \cos x + C$ 
C)  $\frac { 1 } { 8 } [ \sin 2 x - 2 \cos 2 x ] + C$ 
D)  $\frac { 1 } { 8 } [ \sin 8 x - 8 \cos 8 x ] + C$ 
E)  $\text { none of these }$ 
A)  $- \frac { 1 } { 2 } \cos x \sin x + \frac { 1 } { 2 } x + C$ 
B)  $- \frac { 1 } { 2 } x \sin ^ { 2 } x + \frac { 1 } { 2 } \cos x + C$ 
C)  $\frac { 1 } { 8 } [ \sin 2 x - 2 \cos 2 x ] + C$ 
D)  $\frac { 1 } { 8 } [ \sin 8 x - 8 \cos 8 x ] + C$ 
E)  $\text { none of these }$

Question 3

Use Euler's method with step size 0.1 to estimate $y ( 0.5 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { \prime } = y ( x + 2 ) , y ( 0 ) = 1$

Accepted Answer

The answer of Use Euler's method with step size 0.1...

Question 4

Find $f _ { x y }$ for the function $f ( x , y ) = 5 x ^ { 3 } y - 6 x y ^ { 2 }$ .

Accepted Answer

A)  $f _ { x y } = 15 x ^ { 2 } + 12 y$ 
B)  $f _ { x y } = 15 x ^ { 2 } - 12 y$ 
C)  $f _ { x y } = 24 x ^ { 2 } - 12 y$ 
D)  $f _ { x y } = 12 x ^ { 2 } - 15 y$ 
E)  $f _ { x y } = 12 x ^ { 2 } + 15 y$ 
A)  $f _ { x y } = 15 x ^ { 2 } + 12 y$ 
B)  $f _ { x y } = 15 x ^ { 2 } - 12 y$ 
C)  $f _ { x y } = 24 x ^ { 2 } - 12 y$ 
D)  $f _ { x y } = 12 x ^ { 2 } - 15 y$ 
E)  $f _ { x y } = 12 x ^ { 2 } + 15 y$

Question 5

Evaluate the line integral. $\int _ { c } y z \cos x d s$ $C : x = t , y = 4 \cos t , z = 4 \sin t \quad 0 \leq t \leq \pi$

Accepted Answer

A)  $\frac { 4 \sqrt { 5 } } { 3 }$ 
B)  $\frac { 32 \sqrt { 5 } } { 3 }$ 
C)  $\frac { 2 \sqrt { 2 } } { 3 }$ 
D)  $\frac { 32 \sqrt { 5 } } { 5 }$ 
E)  $\frac { \sqrt { 5 } } { 3 }$ 
A)  $\frac { 4 \sqrt { 5 } } { 3 }$ 
B)  $\frac { 32 \sqrt { 5 } } { 3 }$ 
C)  $\frac { 2 \sqrt { 2 } } { 3 }$ 
D)  $\frac { 32 \sqrt { 5 } } { 5 }$ 
E)  $\frac { \sqrt { 5 } } { 3 }$

Question 6

Find the length of the curve. $x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } , 1 \leq y \leq 2$

Accepted Answer

A)  25.05 $5$ 
B)  13.05 
C)  36.05 
D)  $2.0625$ 
E)  None of these 
A)  25.05 $5$ 
B)  13.05 
C)  36.05 
D)  $2.0625$ 
E)  None of these

Question 7

Find the length of the curve. $y = 2 \ln \left( \sin \frac { x } { 2 } \right)$ , $\frac { \pi } { 6 } \leq x \leq \pi$

Accepted Answer

The answer of Find the length of the curve. \(y...

Question 8

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \sin x , y ( 7 \pi ) = 0$

Accepted Answer

A)  $y = - x \sin x + 7 x$ 
B)  $y = x \sin x - 7 x$ 
C)  $y = 7 x \cos x - \sin x$ 
D)  $y = 7 x \sin x$ 
E)  $y = - x \cos x - x$ 
A)  $y = - x \sin x + 7 x$ 
B)  $y = x \sin x - 7 x$ 
C)  $y = 7 x \cos x - \sin x$ 
D)  $y = 7 x \sin x$ 
E)  $y = - x \cos x - x$

Question 9

If $f ( x ) = \sqrt { x } - 3,1 \leq x \leq 6$ , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

Accepted Answer

The answer of If \(f ( x ) = \sqrt...

Question 10

The top of a ladder slides down a vertical wall at a rate of 0.1m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s .
How long is the ladder?

Accepted Answer

The answer of The top of a ladder slides down...

Question 11

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

Accepted Answer

A)  $4.73$ 
B)  $$2.73$$ 
C)  $3.73$ 
D)  $6.73$ 
E)  $5.73$ 
A)  $4.73$ 
B)  $$2.73$$ 
C)  $3.73$ 
D)  $6.73$ 
E)  $5.73$

Question 12

A piece of wire $10$ m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

Accepted Answer

A)  $17.0489$ m 
B)  $3.3496$ m 
C)  $6.3496$ m 
D)  $5.3496$ m 
E)  $4.3496$ m 
A)  $17.0489$ m 
B)  $3.3496$ m 
C)  $6.3496$ m 
D)  $5.3496$ m 
E)  $4.3496$ m

Question 13

Find the volume of the given solid.
Under the paraboloid $z = 3 x ^ { 2 } + 9 y ^ { 2 }$ and above the rectangle $R = [ 0,3 ] \times [ 1,5 ]$ .

Accepted Answer

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

Question 14

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. $P ( 3,0,0 ) , Q ( 5,6,0 ) , R ( 0,6,3 )$

Accepted Answer

A)  $\mathbf { i } - 6 \mathbf { j } + 5 \mathbf { k }$ 
B)  $\mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }$ 
C)  $18 \mathbf { i } - 6 \mathbf { j } + 30 \mathbf { k }$ 
D)  $- 5 \mathbf { i } + \mathbf { j } + 6 \mathbf { k }$ 
E)  None of these 
A)  $\mathbf { i } - 6 \mathbf { j } + 5 \mathbf { k }$ 
B)  $\mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }$ 
C)  $18 \mathbf { i } - 6 \mathbf { j } + 30 \mathbf { k }$ 
D)  $- 5 \mathbf { i } + \mathbf { j } + 6 \mathbf { k }$ 
E)  None of these

Question 15

The height (in meters) of a projectile shot vertically upward from a point $2.5$ m above ground level with an initial velocity of 25.48 m/s is $h = 2.5 + 25.48 t - 4.9 t ^ { 2 }$ after t seconds. 
a. When does the projectile reach its maximum height?
b. What is the maximum height?

Accepted Answer

A)  $$2.8 \mathrm {~s}$$
$34.428 \mathrm {~m}$ 
B)  $2 \mathrm {~s}$
$32.86 \mathrm {~m}$ 
C)  $2.38$
$34.183 \mathrm {~m}$ 
D)  $$2.4 s$$
$34.428 \mathrm {~m}$ 
E)  $$2.6 \mathrm {~s}$$
$35.624 \mathrm {~m}$ 
A)  $$2.8 \mathrm {~s}$$
$34.428 \mathrm {~m}$ 
B)  $2 \mathrm {~s}$
$32.86 \mathrm {~m}$ 
C)  $2.38$
$34.183 \mathrm {~m}$ 
D)  $$2.4 s$$
$34.428 \mathrm {~m}$ 
E)  $$2.6 \mathrm {~s}$$
$35.624 \mathrm {~m}$

Question 16

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $y \sin 7 x = x \cos 7 y , \left( \frac { \pi } { 7 } , \frac { \pi } { 14 } \right)$

Accepted Answer

A)  $y = \frac { x } { 7 } + \frac { \pi } { 14 }$ 
B)  $y = \frac { x } { 7 }$ 
C)  $y = - \frac { x } { 2 } + \frac { \pi } { 2 }$ 
D)  $y = \frac { x } { 14 }$ 
E)  $y = 2 x - \frac { 3 \pi } { 7 }$ 
A)  $y = \frac { x } { 7 } + \frac { \pi } { 14 }$ 
B)  $y = \frac { x } { 7 }$ 
C)  $y = - \frac { x } { 2 } + \frac { \pi } { 2 }$ 
D)  $y = \frac { x } { 14 }$ 
E)  $y = 2 x - \frac { 3 \pi } { 7 }$

Question 17

Find all the second partial derivatives of $f ( x , y ) = 2 x ^ { 3 } y - 6 x y ^ { 2 }$

Accepted Answer

None

Question 18

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 6 } t ^ { 6 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$ .

Accepted Answer

The answer of Find the unit tangent vector for the...

Question 19

Find the average value of the function $f ( t ) = 6 t \sin \left( t ^ { 2 } \right)$ on the interval $[ 0,20 ]$ . Round your answer to 3 decimal places.

Accepted Answer

A)  0.288 
B)  6.228 
C)  0.3 
D)  $0.228$ 
E)  12 
A)  0.288 
B)  6.228 
C)  0.3 
D)  $0.228$ 
E)  12

Question 20

Which equation does the function $y = e ^ { - 3 t }$ satisfy?

Accepted Answer

A)  $y ^ { \prime \prime } - y ^ { \prime } + 12 y = 0$ 
B)  $y ^ { \prime \prime } + y ^ { \prime } - 12 y = 0$ 
C)  $y ^ { \prime \prime } + y ^ { \prime } + 12 y = 0$ 
D)  $y ^ { \prime \prime } - 3 y ^ { \prime } + 12 y = 0$ 
E)  $y ^ { \prime \prime } - y ^ { \prime } - 12 y = 0$ 
A)  $y ^ { \prime \prime } - y ^ { \prime } + 12 y = 0$ 
B)  $y ^ { \prime \prime } + y ^ { \prime } - 12 y = 0$ 
C)  $y ^ { \prime \prime } + y ^ { \prime } + 12 y = 0$ 
D)  $y ^ { \prime \prime } - 3 y ^ { \prime } + 12 y = 0$ 
E)  $y ^ { \prime \prime } - y ^ { \prime } - 12 y = 0$

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. $f ( x ) = 2 \sqrt { x }$ , $[ 0,9 ]$

Evaluate the integral. $\int x \sin x \cos x d x$

Use Euler's method with step size 0.1 to estimate $y ( 0.5 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { \prime } = y ( x + 2 ) , y ( 0 ) = 1$

Find $f _ { x y }$ for the function $f ( x , y ) = 5 x ^ { 3 } y - 6 x y ^ { 2 }$ .

Evaluate the line integral. $\int _ { c } y z \cos x d s$ $C : x = t , y = 4 \cos t , z = 4 \sin t \quad 0 \leq t \leq \pi$

Find the length of the curve. $x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } , 1 \leq y \leq 2$

Find the length of the curve. $y = 2 \ln \left( \sin \frac { x } { 2 } \right)$ , $\frac { \pi } { 6 } \leq x \leq \pi$

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \sin x , y ( 7 \pi ) = 0$

If $f ( x ) = \sqrt { x } - 3,1 \leq x \leq 6$ , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

The top of a ladder slides down a vertical wall at a rate of 0.1m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?

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

A piece of wire $10$ m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

Find the volume of the given solid. Under the paraboloid $z = 3 x ^ { 2 } + 9 y ^ { 2 }$ and above the rectangle $R = [ 0,3 ] \times [ 1,5 ]$ .

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. $P ( 3,0,0 ) , Q ( 5,6,0 ) , R ( 0,6,3 )$

The height (in meters) of a projectile shot vertically upward from a point $2.5$ m above ground level with an initial velocity of 25.48 m/s is $h = 2.5 + 25.48 t - 4.9 t ^ { 2 }$ after t seconds. a. When does the projectile reach its maximum height? b. What is the maximum height?

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $y \sin 7 x = x \cos 7 y , \left( \frac { \pi } { 7 } , \frac { \pi } { 14 } \right)$

Find all the second partial derivatives of $f ( x , y ) = 2 x ^ { 3 } y - 6 x y ^ { 2 }$

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 6 } t ^ { 6 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$ .

Find the average value of the function $f ( t ) = 6 t \sin \left( t ^ { 2 } \right)$ on the interval $[ 0,20 ]$ . Round your answer to 3 decimal places.

Which equation does the function $y = e ^ { - 3 t }$ satisfy?

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Exam 18: Final Exam

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. f(x)=2xf ( x ) = 2 \sqrt { x }f(x)=2x​ , [0,9][ 0,9 ][0,9]

Evaluate the integral. ∫xsin⁡xcos⁡xdx\int x \sin x \cos x d x∫xsinxcosxdx

Use Euler's method with step size 0.1 to estimate y(0.5)y ( 0.5 )y(0.5) , where y(x)y ( x )y(x) is the solution of the initial-value problem. Round your answer to four decimal places. y′=y(x+2),y(0)=1y ^ { \prime } = y ( x + 2 ) , y ( 0 ) = 1y′=y(x+2),y(0)=1

Find fxyf _ { x y }fxy​ for the function f(x,y)=5x3y−6xy2f ( x , y ) = 5 x ^ { 3 } y - 6 x y ^ { 2 }f(x,y)=5x3y−6xy2 .

Evaluate the line integral. ∫cyzcos⁡xds\int _ { c } y z \cos x d s∫c​yzcosxds C:x=t,y=4cos⁡t,z=4sin⁡t0≤t≤πC : x = t , y = 4 \cos t , z = 4 \sin t \quad 0 \leq t \leq \piC:x=t,y=4cost,z=4sint0≤t≤π

Find the length of the curve. x=y48+14y2,1≤y≤2x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } , 1 \leq y \leq 2x=8y4​+4y21​,1≤y≤2

Find the length of the curve. y=2ln⁡(sin⁡x2)y = 2 \ln \left( \sin \frac { x } { 2 } \right)y=2ln(sin2x​) , π6≤x≤π\frac { \pi } { 6 } \leq x \leq \pi6π​≤x≤π

Solve the initial-value problem. xy′=y+x2sin⁡x,y(7π)=0x y ^ { \prime } = y + x ^ { 2 } \sin x , y ( 7 \pi ) = 0xy′=y+x2sinx,y(7π)=0

If f(x)=x−3,1≤x≤6f ( x ) = \sqrt { x } - 3,1 \leq x \leq 6f(x)=x​−3,1≤x≤6 , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

The top of a ladder slides down a vertical wall at a rate of 0.1m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?

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

A piece of wire 101010 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

Find the volume of the given solid. Under the paraboloid z=3x2+9y2z = 3 x ^ { 2 } + 9 y ^ { 2 }z=3x2+9y2 and above the rectangle R=[0,3]×[1,5]R = [ 0,3 ] \times [ 1,5 ]R=[0,3]×[1,5] .

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. P(3,0,0),Q(5,6,0),R(0,6,3)P ( 3,0,0 ) , Q ( 5,6,0 ) , R ( 0,6,3 )P(3,0,0),Q(5,6,0),R(0,6,3)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. ysin⁡7x=xcos⁡7y,(π7,π14)y \sin 7 x = x \cos 7 y , \left( \frac { \pi } { 7 } , \frac { \pi } { 14 } \right)ysin7x=xcos7y,(7π​,14π​)

Find all the second partial derivatives of f(x,y)=2x3y−6xy2f ( x , y ) = 2 x ^ { 3 } y - 6 x y ^ { 2 }f(x,y)=2x3y−6xy2

Find the unit tangent vector for the curve given by r(t)=⟨16t6,13t3,t⟩\mathbf { r } ( t ) = \left\langle \frac { 1 } { 6 } t ^ { 6 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangler(t)=⟨61​t6,31​t3,t⟩ .

Find the average value of the function f(t)=6tsin⁡(t2)f ( t ) = 6 t \sin \left( t ^ { 2 } \right)f(t)=6tsin(t2) on the interval [0,20][ 0,20 ][0,20] . Round your answer to 3 decimal places.

Which equation does the function y=e−3ty = e ^ { - 3 t }y=e−3t satisfy?

Functions and Limits

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Further Applications of Integration

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Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. $f ( x ) = 2 \sqrt { x }$ , $[ 0,9 ]$

Evaluate the integral. $\int x \sin x \cos x d x$

Use Euler's method with step size 0.1 to estimate $y ( 0.5 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { \prime } = y ( x + 2 ) , y ( 0 ) = 1$

Find $f _ { x y }$ for the function $f ( x , y ) = 5 x ^ { 3 } y - 6 x y ^ { 2 }$ .

Evaluate the line integral. $\int _ { c } y z \cos x d s$ $C : x = t , y = 4 \cos t , z = 4 \sin t \quad 0 \leq t \leq \pi$

Find the length of the curve. $x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } , 1 \leq y \leq 2$

Find the length of the curve. $y = 2 \ln \left( \sin \frac { x } { 2 } \right)$ , $\frac { \pi } { 6 } \leq x \leq \pi$

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \sin x , y ( 7 \pi ) = 0$

If $f ( x ) = \sqrt { x } - 3,1 \leq x \leq 6$ , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

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

A piece of wire $10$ m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

Find the volume of the given solid. Under the paraboloid $z = 3 x ^ { 2 } + 9 y ^ { 2 }$ and above the rectangle $R = [ 0,3 ] \times [ 1,5 ]$ .

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. $P ( 3,0,0 ) , Q ( 5,6,0 ) , R ( 0,6,3 )$

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $y \sin 7 x = x \cos 7 y , \left( \frac { \pi } { 7 } , \frac { \pi } { 14 } \right)$

Find all the second partial derivatives of $f ( x , y ) = 2 x ^ { 3 } y - 6 x y ^ { 2 }$

Find the unit tangent vector for the curve given by $\mathbf { r } ( t ) = \left\langle \frac { 1 } { 6 } t ^ { 6 } , \frac { 1 } { 3 } t ^ { 3 } , t \right\rangle$ .

Find the average value of the function $f ( t ) = 6 t \sin \left( t ^ { 2 } \right)$ on the interval $[ 0,20 ]$ . Round your answer to 3 decimal places.

Which equation does the function $y = e ^ { - 3 t }$ satisfy?