Question 1

Suppose that $F ( x ) = f ( g ( x ) )$ and $g ( 14 ) = 2 , g ^ { \prime } ( 14 ) = 4 , f ^ { \prime } ( 14 ) = 15$, and $f ^ { \prime } ( 2 ) = 13$.
Find $F ^ { \prime } ( 14 )$.

Accepted Answer

The answer of Suppose that \(F ( x ) =...

Question 2

Find the derivative of the function.
$$f ( x ) = 0.2 x ^ { - 1.7 }$$

Accepted Answer

A)  $- \frac { 0.34 } { x ^ { 0.2 } }$ 
B)  $- \frac { 0.34 } { x ^ { 2.7 } }$ 
C)  $- 0.34 x ^ { 2.3 }$ 
D)  $- 0.34 x ^ { 0.2 }$ 
A)  $- \frac { 0.34 } { x ^ { 0.2 } }$ 
B)  $- \frac { 0.34 } { x ^ { 2.7 } }$ 
C)  $- 0.34 x ^ { 2.3 }$ 
D)  $- 0.34 x ^ { 0.2 }$

Question 3

The position function of a particle is given by
$s = t ^ { 3 } - 10.5 t ^ { 2 } - 2 t , t \leq 0$
When does the particle reach a velocity of $22 \mathrm {~m} / \mathrm { s }$ ?

Accepted Answer

The answer of The position function of a particle is...

Question 4

A car leaves an intersection traveling west. Its position $4 \mathrm { sec }$ later is $26 \mathrm { ft }$ from the intersection. At the same time, another car leaves the same intersection heading north so that its position 4 sec later is $26 \mathrm { ft }$ from the intersection. If the speeds of the cars at that instant of time are $12 \mathrm { ft } / \mathrm { sec }$ and 10 $\mathrm { ft } / \mathrm { sec }$, respectively, find the rate at which the distance between the two cars is changing. Round to the nearest tenth if necessary.
Select the correct answer.

Accepted Answer

A)  $\quad 15.6 \mathrm { ft } / \mathrm { sec }$ 
B)  $3.7 \mathrm { ft } / \mathrm { sec }$ 
C)  $3.1 \mathrm { ft } / \mathrm { sec }$ 
D)  $36.8 \mathrm { ft } / \mathrm { sec }$ 
A)  $\quad 15.6 \mathrm { ft } / \mathrm { sec }$ 
B)  $3.7 \mathrm { ft } / \mathrm { sec }$ 
C)  $3.1 \mathrm { ft } / \mathrm { sec }$ 
D)  $36.8 \mathrm { ft } / \mathrm { sec }$

Question 5

A plane flying horizontally at an altitude of $1 \mathrm { mi }$ and a speed of $550 \mathrm { mi } / \mathrm { h }$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.
Select the correct answer.

Accepted Answer

A)  $\approx 476 \mathrm { mi } / \mathrm { h }$ 
B)  $\approx 670 \mathrm { mi } / \mathrm { h }$ 
C)  $\approx 455 \mathrm { mi } \mathrm { h }$ 
D)  $\approx 570 \mathrm { mi } / \mathrm { h }$ 
E)  $\approx 495 \mathrm { mi } / \mathrm { h }$ 
A)  $\approx 476 \mathrm { mi } / \mathrm { h }$ 
B)  $\approx 670 \mathrm { mi } / \mathrm { h }$ 
C)  $\approx 455 \mathrm { mi } \mathrm { h }$ 
D)  $\approx 570 \mathrm { mi } / \mathrm { h }$ 
E)  $\approx 495 \mathrm { mi } / \mathrm { h }$

Question 6

Find the differential of the function at the indicated number.
$$f ( x ) = 13 \sin x + 4 \cos x ; \quad x = \frac { \pi } { 4 }$$

Accepted Answer

A)  $\frac { 9 \sqrt { 2 } } { 2 } d x$ 
B)  $- \frac { 17 \sqrt { 2 } } { 2 } d x$ 
C)  $\frac { 17 \sqrt { 2 } } { 2 } d x$ 
D)  $- \frac { 9 \sqrt { 2 } } { 2 } d x$ 
A)  $\frac { 9 \sqrt { 2 } } { 2 } d x$ 
B)  $- \frac { 17 \sqrt { 2 } } { 2 } d x$ 
C)  $\frac { 17 \sqrt { 2 } } { 2 } d x$ 
D)  $- \frac { 9 \sqrt { 2 } } { 2 } d x$

Question 7

Differentiate.
$$g ( x ) = 2 \sec x + \tan x$$

Accepted Answer

The answer of Differentiate.
\[g ( x ) = 2 \sec...

Question 8

Find $y ^ { \prime }$ by implicit differentiation.
$10 \cos x \sin y = 16$

Accepted Answer

The answer of Find $y ^ { \prime }$ by...

Question 9

Two chemicals react to form another chemical. Suppose that the amount of chemical formed in time $t$ (in hours) is given by
$$x ( t ) = \frac { 11 \left[ 1 - \left( \frac { 2 } { 3 } \right) ^ { 3 t } \right] } { 1 - \frac { 1 } { 4 } \left( \frac { 2 } { 3 } \right) ^ { 3 t } }$$
where $x ( t )$ is measured in pounds.
a. Find the rate at which the chemical is formed when $t = 4$. Round to two decimal places.
b. How many pounds of the chemical are formed eventually?

Accepted Answer

The answer of Two chemicals react to form another chemical....

Question 10

Find the rate of change of $y$ with respect to $x$ at the given values of $x$ and $y$.
$$2 x y ^ { 2 } - 5 x ^ { 2 } y + 192 = 0 ; \quad x = 4 , y = 4$$

Accepted Answer

The answer of Find the rate of change of $y$...

Question 11

A plane flying horizontally at an altitude of $1 \mathrm { mi }$ and a speed of $550 \mathrm { mi } / \mathrm { h }$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is $2 \mathrm { mi }$ away from the station.

Accepted Answer

A)  $\approx 476 \mathrm { mi } / \mathrm { h }$ 
B)  $\approx 670 \mathrm { mi } / \mathrm { h }$ 
C)  $\approx 455 \mathrm { mi } / \mathrm { h }$ 
D)  $\approx 570 \mathrm { mi } / \mathrm { h }$ 
E)  $\approx 495 \mathrm { mih }$ 
A)  $\approx 476 \mathrm { mi } / \mathrm { h }$ 
B)  $\approx 670 \mathrm { mi } / \mathrm { h }$ 
C)  $\approx 455 \mathrm { mi } / \mathrm { h }$ 
D)  $\approx 570 \mathrm { mi } / \mathrm { h }$ 
E)  $\approx 495 \mathrm { mih }$

Question 12

A turkey is removed from the oven when its temperature reaches $175 ^ { \circ } \mathrm { F }$ and is placed on a table in a room where the temperature is $70 ^ { \circ } \mathrm { F }$. After 10 minutes the temperature of the turkey is 161 ${ } ^ { \circ } \mathrm { F }$ and after 20 minutes it is $149 ^ { \circ } \mathrm { F }$. Use a linear approximation to predict the temperature of the turkey after 30 minutes.

Accepted Answer

The answer of A turkey is removed from the oven...

Question 13

The altitude of a triangle is increasing at a rate of $1 \mathrm {~cm} / \mathrm { min }$ while the area of the triangle is increasing at a rate of $2 \mathrm {~cm} ^ { 2 } / \mathrm { min }$. At what rate is the base of the triangle changing when the altitude is $10 \mathrm {~cm}$ and the area is $100 \mathrm {~cm} ^ { 2 }$.

Accepted Answer

The answer of The altitude of a triangle is increasing...

Question 14

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$.
$$x ^ { 7 } - y ^ { 7 } = 1$$ Calculate $y ^ { t }$.
$$x y ^ { 3 } + x ^ { 3 } y = x + 3 y$$

Accepted Answer

The answer of Find \(\frac { d ^ { 2...

Question 15

The quantity $Q$ of charge in coulombs $C$ that has passed through a point in a wire up to time $t$ (measured in seconds) is given by
$$Q ( t ) = t ^ { 3 } - 3 t ^ { 2 } + 4 t + 3$$
Find the current when $t = 1 \mathrm {~s}$.

Accepted Answer

A)  24 
B)  15 
C)  18 
D)  26 
E)  1 
A)  24 
B)  15 
C)  18 
D)  26 
E)  1

Question 16

$$\text { If } g ( x ) = \sqrt { 8 - 7 x } \text {, find the domain of } g ^ { \prime } ( x ) \text {. }$$

Accepted Answer

The answer of \[\text { If } g ( x...

Question 17

$$\text { Find an equation of the tangent line to the graph of } f ( x ) = 2 x ^ { 2 } - 7 \text { at the point } ( 3,11 ) \text {. }$$

Accepted Answer

The answer of \[\text { Find an equation of the...

Question 18

Find $\frac { d y } { d x }$ by implicit differentiation.
$$8 \sqrt { x } + \sqrt { y } = 8$$

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 19

A plane flying horizontally at an altitude of $1 \mathrm { mi }$ and a speed of $550 \mathrm { mi } / \mathrm { h }$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is $2 \mathrm { mi }$ away from the station.

Accepted Answer

The answer of A plane flying horizontally at an altitude...

Question 20

Determine the values of $x$ for which the given linear approximation is accurate to within $0.07$ at $a = 0$.
$\tan x \approx x$

Accepted Answer

A)  $- 0.19 < x < 0.28$ 
B)  $- 0.57 < x < 0.57$ 
C)  $\quad 0.06 < x < 0.68$ 
D)  $- 1.04 < x < 1.55$ 
E)  $- 0.71 < x < 0.48$ 
A)  $- 0.19 < x < 0.28$ 
B)  $- 0.57 < x < 0.57$ 
C)  $\quad 0.06 < x < 0.68$ 
D)  $- 1.04 < x < 1.55$ 
E)  $- 0.71 < x < 0.48$

Suppose that $F ( x ) = f ( g ( x ) )$ and $g ( 14 ) = 2 , g ^ { \prime } ( 14 ) = 4 , f ^ { \prime } ( 14 ) = 15$ , and $f ^ { \prime } ( 2 ) = 13$ . Find $F ^ { \prime } ( 14 )$ .

Find the derivative of the function. $f ( x ) = 0.2 x ^ { - 1.7 }$

The position function of a particle is given by $s = t ^ { 3 } - 10.5 t ^ { 2 } - 2 t , t \leq 0$ When does the particle reach a velocity of $22 \mathrm {~m} / \mathrm { s }$ ?

A plane flying horizontally at an altitude of $1 \mathrm { mi }$ and a speed of $550 \mathrm { mi } / \mathrm { h }$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station. Select the correct answer.

Find the differential of the function at the indicated number. $f ( x ) = 13 \sin x + 4 \cos x ; \quad x = \frac { \pi } { 4 }$

Differentiate. $g ( x ) = 2 \sec x + \tan x$

Find $y ^ { \prime }$ by implicit differentiation. $10 \cos x \sin y = 16$

Find the rate of change of $y$ with respect to $x$ at the given values of $x$ and $y$ . $2 x y ^ { 2 } - 5 x ^ { 2 } y + 192 = 0 ; \quad x = 4 , y = 4$

A plane flying horizontally at an altitude of $1 \mathrm { mi }$ and a speed of $550 \mathrm { mi } / \mathrm { h }$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is $2 \mathrm { mi }$ away from the station.

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$ . $x ^ { 7 } - y ^ { 7 } = 1$ Calculate $y ^ { t }$ . $x y ^ { 3 } + x ^ { 3 } y = x + 3 y$

The quantity $Q$ of charge in coulombs $C$ that has passed through a point in a wire up to time $t$ (measured in seconds) is given by $Q ( t ) = t ^ { 3 } - 3 t ^ { 2 } + 4 t + 3$ Find the current when $t = 1 \mathrm {~s}$ .

$\text { If } g ( x ) = \sqrt { 8 - 7 x } \text {, find the domain of } g ^ { \prime } ( x ) \text {. }$

$\text { Find an equation of the tangent line to the graph of } f ( x ) = 2 x ^ { 2 } - 7 \text { at the point } ( 3,11 ) \text {. }$

Find $\frac { d y } { d x }$ by implicit differentiation. $8 \sqrt { x } + \sqrt { y } = 8$

A plane flying horizontally at an altitude of $1 \mathrm { mi }$ and a speed of $550 \mathrm { mi } / \mathrm { h }$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is $2 \mathrm { mi }$ away from the station.

Determine the values of $x$ for which the given linear approximation is accurate to within $0.07$ at $a = 0$ . $\tan x \approx x$

Functions and Models

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Exam 2: Limits and Derivatives

Suppose that F(x)=f(g(x))F ( x ) = f ( g ( x ) )F(x)=f(g(x)) and g(14)=2,g′(14)=4,f′(14)=15g ( 14 ) = 2 , g ^ { \prime } ( 14 ) = 4 , f ^ { \prime } ( 14 ) = 15g(14)=2,g′(14)=4,f′(14)=15 , and f′(2)=13f ^ { \prime } ( 2 ) = 13f′(2)=13 . Find F′(14)F ^ { \prime } ( 14 )F′(14) .

Find the derivative of the function. f(x)=0.2x−1.7f ( x ) = 0.2 x ^ { - 1.7 }f(x)=0.2x−1.7

The position function of a particle is given by s=t3−10.5t2−2t,t≤0s = t ^ { 3 } - 10.5 t ^ { 2 } - 2 t , t \leq 0s=t3−10.5t2−2t,t≤0 When does the particle reach a velocity of 22 m/s22 \mathrm {~m} / \mathrm { s }22 m/s ?

Find the differential of the function at the indicated number. f(x)=13sin⁡x+4cos⁡x;x=π4f ( x ) = 13 \sin x + 4 \cos x ; \quad x = \frac { \pi } { 4 }f(x)=13sinx+4cosx;x=4π​

Differentiate. g(x)=2sec⁡x+tan⁡xg ( x ) = 2 \sec x + \tan xg(x)=2secx+tanx

Find y′y ^ { \prime }y′ by implicit differentiation. 10cos⁡xsin⁡y=1610 \cos x \sin y = 1610cosxsiny=16

Find the rate of change of yyy with respect to xxx at the given values of xxx and yyy . 2xy2−5x2y+192=0;x=4,y=42 x y ^ { 2 } - 5 x ^ { 2 } y + 192 = 0 ; \quad x = 4 , y = 42xy2−5x2y+192=0;x=4,y=4

Find d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ in terms of xxx and yyy . x7−y7=1x ^ { 7 } - y ^ { 7 } = 1x7−y7=1 Calculate yty ^ { t }yt . xy3+x3y=x+3yx y ^ { 3 } + x ^ { 3 } y = x + 3 yxy3+x3y=x+3y

The quantity QQQ of charge in coulombs CCC that has passed through a point in a wire up to time ttt (measured in seconds) is given by Q(t)=t3−3t2+4t+3Q ( t ) = t ^ { 3 } - 3 t ^ { 2 } + 4 t + 3Q(t)=t3−3t2+4t+3 Find the current when t=1 st = 1 \mathrm {~s}t=1 s .

If g(x)=8−7x, find the domain of g′(x). \text { If } g ( x ) = \sqrt { 8 - 7 x } \text {, find the domain of } g ^ { \prime } ( x ) \text {. } If g(x)=8−7x​, find the domain of g′(x).

Find dydx\frac { d y } { d x }dxdy​ by implicit differentiation. 8x+y=88 \sqrt { x } + \sqrt { y } = 88x​+y​=8

Determine the values of xxx for which the given linear approximation is accurate to within 0.070.070.07 at a=0a = 0a=0 . tan⁡x≈x\tan x \approx xtanx≈x

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Applications of Integration

Techniques of Integration

Further Applications of Integration

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Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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Suppose that $F ( x ) = f ( g ( x ) )$ and $g ( 14 ) = 2 , g ^ { \prime } ( 14 ) = 4 , f ^ { \prime } ( 14 ) = 15$ , and $f ^ { \prime } ( 2 ) = 13$ . Find $F ^ { \prime } ( 14 )$ .

Find the derivative of the function. $f ( x ) = 0.2 x ^ { - 1.7 }$

The position function of a particle is given by $s = t ^ { 3 } - 10.5 t ^ { 2 } - 2 t , t \leq 0$ When does the particle reach a velocity of $22 \mathrm {~m} / \mathrm { s }$ ?

Find the differential of the function at the indicated number. $f ( x ) = 13 \sin x + 4 \cos x ; \quad x = \frac { \pi } { 4 }$

Differentiate. $g ( x ) = 2 \sec x + \tan x$

Find $y ^ { \prime }$ by implicit differentiation. $10 \cos x \sin y = 16$

Find the rate of change of $y$ with respect to $x$ at the given values of $x$ and $y$ . $2 x y ^ { 2 } - 5 x ^ { 2 } y + 192 = 0 ; \quad x = 4 , y = 4$

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$ . $x ^ { 7 } - y ^ { 7 } = 1$ Calculate $y ^ { t }$ . $x y ^ { 3 } + x ^ { 3 } y = x + 3 y$

The quantity $Q$ of charge in coulombs $C$ that has passed through a point in a wire up to time $t$ (measured in seconds) is given by $Q ( t ) = t ^ { 3 } - 3 t ^ { 2 } + 4 t + 3$ Find the current when $t = 1 \mathrm {~s}$ .

$\text { If } g ( x ) = \sqrt { 8 - 7 x } \text {, find the domain of } g ^ { \prime } ( x ) \text {. }$

Find $\frac { d y } { d x }$ by implicit differentiation. $8 \sqrt { x } + \sqrt { y } = 8$

Determine the values of $x$ for which the given linear approximation is accurate to within $0.07$ at $a = 0$ . $\tan x \approx x$