Question 1

Calculate y'. $y=5 \ln \left(x^{2} e^{x}\right)$

Accepted Answer

The answer of Calculate y'. $y=5 \ln \left(x^{2} e^{x}\right)$...

Question 2

Use logarithmic differentiation to find the derivative of the function. $y=\sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$

Accepted Answer

A)  $y^{\prime}=-\frac{9 x}{\left(x^{4}-1\right)} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$ 
B)  $y^{\prime}=\frac{9 x}{4 x^{4}-1} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$ 
C)  $y^{\prime}=-\frac{36 x}{x^{4}-1} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$ 
D)  $y^{\prime}=-\frac{36 x}{x^{4}-1}$ 
E)  $y^{\prime}=-\frac{4 x}{9\left(x^{4}-1\right)} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$ 
A)  $y^{\prime}=-\frac{9 x}{\left(x^{4}-1\right)} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$ 
B)  $y^{\prime}=\frac{9 x}{4 x^{4}-1} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$ 
C)  $y^{\prime}=-\frac{36 x}{x^{4}-1} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$ 
D)  $y^{\prime}=-\frac{36 x}{x^{4}-1}$ 
E)  $y^{\prime}=-\frac{4 x}{9\left(x^{4}-1\right)} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$

Question 3

Find equations of the tangent lines to the curve $y=\frac{x-8}{x+8}$ that are parallel to the line $x-y=8$ .

Accepted Answer

A)  $x-y=-18.5$ 
B)  $x-y=-4.5$ 
C)  $x-y=-1.5$ 
D)  $x-y=-12.5$ 
E)  $x-y=-15$ 
A)  $x-y=-18.5$ 
B)  $x-y=-4.5$ 
C)  $x-y=-1.5$ 
D)  $x-y=-12.5$ 
E)  $x-y=-15$

Question 4

Differentiate. $g(x)=9 \sec x+\tan x$

Accepted Answer

The answer of Differentiate. $g(x)=9 \sec x+\tan x$...

Question 5

The mass of the part of a metal rod that lies between its left end and a point x meters to the right is $S=4 x^{2}$ . Find the linear density when x is 3 m.

Accepted Answer

A)  4 
B)  20 
C)  24 
D)  12 
E)  18 
A)  4 
B)  20 
C)  24 
D)  12 
E)  18

Question 6

Use the linear approximation of the function $f(x)=\sqrt{9-x}$ at $a=0$ to approximate the number $\sqrt{9.08}$ .

Accepted Answer

A)  $7.4445$ 
B)  $2.2556$ 
C)  $3.0133$ 
D)  $7.4556$ 
E)  $0.1556$ 
A)  $7.4445$ 
B)  $2.2556$ 
C)  $3.0133$ 
D)  $7.4556$ 
E)  $0.1556$

Question 7

The top of a ladder leaning against a wall is 8 ft above the ground. The slope of the ladder with respect to the ground is -4. What is the length of the ladder?

Accepted Answer

The answer of The top of a ladder leaning against...

Question 8

The equation of motion is given for a particle, where s is in meters and t is in seconds. Find the acceleration after $2.5$ seconds. $s=\sin 2 \pi t$

Accepted Answer

A)  $0 \mathrm{~m} / \mathrm{s}^{2}$ 
B)  $-15.625 ~\pi^{2} \mathrm{~m} / \mathrm{s}^{2}$ 
C)  $-6.25 ~\pi \mathrm{m} / \mathrm{s}^{2}$ 
D)  $15.625 \pi^{2} \mathrm{~m} / \mathrm{s}^{2}$ 
E)  $6.25 \pi \mathrm{m} / \mathrm{s}^{2}$ 
A)  $0 \mathrm{~m} / \mathrm{s}^{2}$ 
B)  $-15.625 ~\pi^{2} \mathrm{~m} / \mathrm{s}^{2}$ 
C)  $-6.25 ~\pi \mathrm{m} / \mathrm{s}^{2}$ 
D)  $15.625 \pi^{2} \mathrm{~m} / \mathrm{s}^{2}$ 
E)  $6.25 \pi \mathrm{m} / \mathrm{s}^{2}$

Question 9

Water flows from a tank of constant cross-sectional area 50 $\mathrm{ft}^{2}$ through an orifice of constant cross-sectional area $\frac{1}{4}$ $\mathrm{ft}^{2}$ located at the bottom of the tank. Initially, the height of the water in the tank was 20 ft, and t sec later it was given by the equation $2 \sqrt{h}+\frac{1}{25} t-2 \sqrt{20}=0 \quad \quad \quad 0 \leq t \leq 50 \sqrt{20}$ How fast was the height of the water decreasing when its height was 2 ft?

Accepted Answer

A)  $100 \sqrt{5}-50 \sqrt{2}$ ft/sec 
B)  $100 \sqrt{5}-50 \sqrt{2}$ ft/sec. 
C)  $\frac{2}{25}$ ft/sec 
D)  $\frac{\sqrt{2}}{25}$ ft/sec 
A)  $100 \sqrt{5}-50 \sqrt{2}$ ft/sec 
B)  $100 \sqrt{5}-50 \sqrt{2}$ ft/sec. 
C)  $\frac{2}{25}$ ft/sec 
D)  $\frac{\sqrt{2}}{25}$ ft/sec

Question 10

If a snowball melts so that its surface area decreases at a rate of $4 \mathrm{~cm}^{2} / \mathrm{min}$ , find the rate at which the diameter decreases when the diameter is 35 cm.

Accepted Answer

The answer of If a snowball melts so that its...

Question 11

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 $t=3 s$ .

Accepted Answer

A)  15 
B)  24 
C)  26 
D)  18 
E)  13 
A)  15 
B)  24 
C)  26 
D)  18 
E)  13

Question 12

Use logarithmic differentiation to find the derivative of the function. $y=x^{x^{2}}$

Accepted Answer

The answer of Use logarithmic differentiation to find the derivative...

Question 13

In an adiabatic process (one in which no heat transfer takes place), the pressure P and volume V of an ideal gas such as oxygen satisfy the equation $p^{5} V^{7}=C$ , where C is a constant. Suppose that at a certain instant of time, the volume of the gas is 2L, the pressure is 100 kPa, and the pressure is decreasing at the rate of 5 kPa/sec. Find the rate at which the volume is changing.

Accepted Answer

A)  14 L/sec 
B)  $C-$ 14 L/sec 
C)  $C-$ $\frac{1}{14}$ L/sec 
D)  $\frac{1}{14}$ L/sec 
A)  14 L/sec 
B)  $C-$ 14 L/sec 
C)  $C-$ $\frac{1}{14}$ L/sec 
D)  $\frac{1}{14}$ L/sec

Question 14

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s how fast is the boat approaching the dock when it is 7 m from the dock? Round the result to the nearest hundredth if necessary.

Accepted Answer

The answer of A boat is pulled into a dock...

Question 15

Find the equation of the tangent line to the given curve at the specified point. $y=8 x e^{x},(0,0)$

Accepted Answer

The answer of Find the equation of the tangent line...

Calculate y'. $y=5 \ln \left(x^{2} e^{x}\right)$

Use logarithmic differentiation to find the derivative of the function. $y=\sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$

Find equations of the tangent lines to the curve $y=\frac{x-8}{x+8}$ that are parallel to the line $x-y=8$ .

Differentiate. $g(x)=9 \sec x+\tan x$

The mass of the part of a metal rod that lies between its left end and a point x meters to the right is $S=4 x^{2}$ . Find the linear density when x is 3 m.

Use the linear approximation of the function $f(x)=\sqrt{9-x}$ at $a=0$ to approximate the number $\sqrt{9.08}$ .

The top of a ladder leaning against a wall is 8 ft above the ground. The slope of the ladder with respect to the ground is -4. What is the length of the ladder?

The equation of motion is given for a particle, where s is in meters and t is in seconds. Find the acceleration after $2.5$ seconds. $s=\sin 2 \pi t$

If a snowball melts so that its surface area decreases at a rate of $4 \mathrm{~cm}^{2} / \mathrm{min}$ , find the rate at which the diameter decreases when the diameter is 35 cm.

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 $t=3 s$ .

Use logarithmic differentiation to find the derivative of the function. $y=x^{x^{2}}$

Find the equation of the tangent line to the given curve at the specified point. $y=8 x e^{x},(0,0)$

Functions and Limits

Derivatives

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 3: Applications of Differentiation

Calculate y'. y=5ln⁡(x2ex)y=5 \ln \left(x^{2} e^{x}\right)y=5ln(x2ex)

Use logarithmic differentiation to find the derivative of the function. y=x2+1x2−19y=\sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}y=9x2−1x2+1​​

Find equations of the tangent lines to the curve y=x−8x+8y=\frac{x-8}{x+8}y=x+8x−8​ that are parallel to the line x−y=8x-y=8x−y=8 .

Differentiate. g(x)=9sec⁡x+tan⁡xg(x)=9 \sec x+\tan xg(x)=9secx+tanx

The mass of the part of a metal rod that lies between its left end and a point x meters to the right is S=4x2S=4 x^{2}S=4x2 . Find the linear density when x is 3 m.

Use the linear approximation of the function f(x)=9−xf(x)=\sqrt{9-x}f(x)=9−x​ at a=0a=0a=0 to approximate the number 9.08\sqrt{9.08}9.08​ .

The top of a ladder leaning against a wall is 8 ft above the ground. The slope of the ladder with respect to the ground is -4. What is the length of the ladder?

The equation of motion is given for a particle, where s is in meters and t is in seconds. Find the acceleration after 2.52.52.5 seconds. s=sin⁡2πts=\sin 2 \pi ts=sin2πt

If a snowball melts so that its surface area decreases at a rate of 4 cm2/min4 \mathrm{~cm}^{2} / \mathrm{min}4 cm2/min , find the rate at which the diameter decreases when the diameter is 35 cm.

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)=t3−3t2+4t+3Q(t)=t^{3}-3 t^{2}+4 t+3Q(t)=t3−3t2+4t+3 . Find the current when t=3st=3 st=3s .

Use logarithmic differentiation to find the derivative of the function. y=xx2y=x^{x^{2}}y=xx2

Find the equation of the tangent line to the given curve at the specified point. y=8xex,(0,0)y=8 x e^{x},(0,0)y=8xex,(0,0)

Functions and Limits

Derivatives

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Calculate y'. $y=5 \ln \left(x^{2} e^{x}\right)$

Use logarithmic differentiation to find the derivative of the function. $y=\sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$

Find equations of the tangent lines to the curve $y=\frac{x-8}{x+8}$ that are parallel to the line $x-y=8$ .

Differentiate. $g(x)=9 \sec x+\tan x$

The mass of the part of a metal rod that lies between its left end and a point x meters to the right is $S=4 x^{2}$ . Find the linear density when x is 3 m.

Use the linear approximation of the function $f(x)=\sqrt{9-x}$ at $a=0$ to approximate the number $\sqrt{9.08}$ .

The equation of motion is given for a particle, where s is in meters and t is in seconds. Find the acceleration after $2.5$ seconds. $s=\sin 2 \pi t$

If a snowball melts so that its surface area decreases at a rate of $4 \mathrm{~cm}^{2} / \mathrm{min}$ , find the rate at which the diameter decreases when the diameter is 35 cm.

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 $t=3 s$ .

Use logarithmic differentiation to find the derivative of the function. $y=x^{x^{2}}$

Find the equation of the tangent line to the given curve at the specified point. $y=8 x e^{x},(0,0)$