ch8

Evaluate the limitlim

x

e^x

x^k

,

wherekis a positive constant.

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2.

[-/1 Points]

DETAILS

OSCALC1 4.8.358.

MY NOTES

ASK YOUR TEACHER

Evaluate the limitlim

x

ln(x)

x^k

,

wherekis a positive constant.

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3.

[-/1 Points]

DETAILS

OSCALC1 4.8.361.

MY NOTES

ASK YOUR TEACHER

Evaluate the limitlim

xa

xa

xⁿaⁿ

,

wherenandaare constants.

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4.

[-/3 Points]

DETAILS

OSCALC1 4.8.362.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Consider the following limit.

lim

x0⁺

x⁴ln(x)

Determine whether you can apply L'Hpital's rule directly. Explain why or why not.

L'Hpital's rule

---Select---

can

cannot

be applied directly because the limit is of the following form.

0

0

1

⁰

0⁰

0

none of the above

How can the limit be altered in order to apply L'Hpital's rule? (Select all that apply.)

The limit can be rewritten using algebraic manipulation.

The limit can be rewritten by taking the logarithm of the limit and using properties of logarithms.

The limit cannot be rewritten in a form where L'Hpital's rule can be applied.

The limit does not need to be rewritten since L'Hpital's rule can be applied directly.

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5.

[-/3 Points]

DETAILS

OSCALC1 4.8.364.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Consider the following limit.

lim

x0

x^2x

Determine whether you can apply L'Hpital's rule directly. Explain why or why not.

L'Hpital's rule

---Select---

can

cannot

be applied directly because the limit is of the following form.

0

0

1

⁰

0⁰

0

none of the above

How can the limit be altered in order to apply L'Hpital's rule? (Select all that apply.)

The limit can be rewritten using algebraic manipulation.

The limit can be rewritten by taking the logarithm of the limit and using properties of logarithms.

The limit cannot be rewritten in a form where L'Hpital's rule can be applied.

The limit does not need to be rewritten since L'Hpital's rule can be applied directly.

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6.

[-/3 Points]

DETAILS

OSCALC1 4.8.366.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Consider the following limit.

lim

x

e^x

x²

Determine whether you can apply L'Hpital's rule directly. Explain why or why not.

L'Hpital's rule

---Select---

can

cannot

be applied directly because the limit is of the following form.

0

0

1

⁰

0⁰

0

none of the above

How can the limit be altered in order to apply L'Hpital's rule? (Select all that apply.)

The limit can be rewritten using algebraic manipulation.

The limit can be rewritten by taking the logarithm of the limit and using properties of logarithms.

The limit cannot be rewritten in a form where L'Hpital's rule can be applied.

The limit does not need to be rewritten since L'Hpital's rule can be applied directly.

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7.

[-/1 Points]

DETAILS

OSCALC1 4.8.367-395A.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the limit with either L'Hpital's rule or previously learned methods.

lim

t

9tln(t)

6+t²

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8.

[-/1 Points]

DETAILS

OSCALC1 4.8.367-395B.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the limit with either L'Hpital's Rule or previously learned methods.

lim

x0

2e^4x2

sin(x)

TutorialAdditional Materials

• eBook

9.

[-/1 Points]

DETAILS

OSCALC1 4.8.367-395C.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the limit with either L'Hpital's Rule or previously learned methods.

lim

x9

9

x+72

81x²

TutorialAdditional Materials

• eBook

10.

[-/1 Points]

DETAILS

OSCALC1 4.8.367-395E.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the limit with either L'Hpital's Rule or previously learned methods.

lim

x0

6x+e^xe^7x

1cos(8x)

TutorialAdditional Materials

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11.

[-/1 Points]

DETAILS

OSCALC1 4.8.370.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the limit with either L'Hpital's rule or previously learned methods.

lim

x/2

cos(x)

2

x

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12.

[-/1 Points]

DETAILS

OSCALC1 4.8.377.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the limit with either L'Hpital's rule or previously learned methods.

lim

x0

e^xx1

5x²

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13.

[-/1 Points]

DETAILS

OSCALC1 4.8.382.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the limit with either L'Hpital's rule or previously learned methods.

lim

x0⁺

x^7x

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14.

[-/1 Points]

DETAILS

OSCALC1 4.8.385.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the limit with either L'Hpital's rule or previously learned methods.

lim

x0⁺

xln(x³)

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• eBook

15.

[-/1 Points]

DETAILS

OSCALC1 4.8.387.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the limit with either L'Hpital's rule or previously learned methods.

lim

x

x⁹e^x

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16.

[-/1 Points]

DETAILS

OSCALC1 4.8.402.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Use a calculator to graph the function and estimate the value of the limit, then use L'Hpital's rule to find the limit directly.

lim

x0

2csc(x)2

x

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17.

[-/1 Points]

DETAILS

OSCALC1 4.8.404.

MY NOTES

ASK YOUR TEACHER

Use a calculator to graph the function and estimate the value of the limit, then use L'Hpital's rule to find the limit directly.

lim

x0⁺

ln(x)

sin(x)

Computex₁

andx₂

using the specified iterative method.

x_{n+ 1}=x_n²1

2

(a)

Start atx₀=0.2.

x₁

=x₂

=

(b)

Start atx₀=2.

x₁

=x₂

=

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2.

[-/4 Points]

DETAILS

OSCALC1 4.9.416.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Computex₁

andx₂

using the specified iterative method. (Round your answers to four decimal places.)

x_{n+ 1}=

x_n

(a)

Start atx₀=0.2.

x₁

=x₂

=

(b)

Start atx₀=3.

x₁

=x₂

=

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3.

[-/4 Points]

DETAILS

OSCALC1 4.9.418.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Computex₁

andx₂

using the specified iterative method.

x_{n+ 1}= 3x_n(1x_n)

(a)

Start atx₀=0.2.

x₁

=x₂

=

(b)

Start atx₀=2.

x₁

=x₂

=

Use the specified method to attempt to solve the equation. If it does not work, explain why it does not work. (Select all that apply.)

Newton's method,x²+7= 0

Newton's method works and the zeros are7 and 7.

Newton's method works and the zeros are2.6458 and 2.6458.

There are no real solutions to the equation.

x₁is always undefined.

Newton's method alternates back and forth between two values.

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6.

[-/1 Points]

DETAILS

OSCALC1 4.9.449.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Use the specified method to attempt to solve the equation. If it does not work, explain why it does not work. (Select all that apply.)

solvingx_{n+ 1}=x_n³

starting atx₀=1

This method works and the zero is 0.

This method works and the zero is 1

x₁is always undefined.

This method alternates back and forth between two values.

There are no real solutions to the equation.

Find the antiderivative of the function. (UseCfor the constant of the antiderivative.)

f(x) =e^x9x⁸+ sin(x)

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2.

[-/1 Points]

DETAILS

OSCALC1 4.10.473.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the antiderivative of the function. (UseCfor the constant of the antiderivative.)

f(x) =x1 +4sin(2x)

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3.

[-/1 Points]

DETAILS

OSCALC1 4.10.474-489A.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the general antiderivative for the function below; assume we have chosen an intervalIon which the function is continuous. The function may require an algebraic manipulation first. (UseCfor the constant of integration.)

f(x) =39

40x^7/8

F(x) =

TutorialAdditional Materials

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4.

[-/1 Points]

DETAILS

OSCALC1 4.10.474-489B.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the general antiderivative for the function below; assume we have chosen an intervalIon which the function is continuous. The function may require an algebraic manipulation first. (UseCfor the constant of integration.)

f(x) =5sec²(x)

F(x) =

TutorialAdditional Materials

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5.

[-/1 Points]

DETAILS

OSCALC1 4.10.474-489C.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the general antiderivative for the function below; assume we have chosen an intervalIon which the function is continuous. The function may require an algebraic manipulation first. (UseCfor the constant of integration.)

f(x) =16sin(x) +4cos(4x)

F(x) =

TutorialAdditional Materials

• eBook

6.

[-/1 Points]

DETAILS

OSCALC1 4.10.474-489D.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the general antiderivative for the function below; assume we have chosen an intervalIon which the function is continuous. The function may require an algebraic manipulation first. (UseCfor the constant of integration. Remember to use absolute values where appropriate.)

f(x) =(6+

x

)²

x

F(x) =

TutorialAdditional Materials

• eBook

7.

[-/1 Points]

DETAILS

OSCALC1 4.10.474-489E.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the general antiderivative for the function below; assume we have chosen an intervalIon which the function is continuous. The function may require an algebraic manipulation first. (UseCfor the constant of integration.)

f(x) =36x

6

x

F(x) =

TutorialAdditional Materials

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8.

[-/1 Points]

DETAILS

OSCALC1 4.10.475.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the antiderivativeF(x)

of the functionf(x).

(UseCfor the constant of the antiderivative.)

f(x) =x+27x²

F(x) =

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9.

[-/1 Points]

DETAILS

OSCALC1 4.10.477.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the antiderivativeF(x)

of the functionf(x).

(UseCfor the constant of the antiderivative.)

f(x) = (

x

)⁵

F(x) =

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10.

[-/1 Points]

DETAILS

OSCALC1 4.10.479.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the antiderivativeF(x)

of the functionf(x).

(UseCfor the constant of the antiderivative.)

f(x) =x^1/3

x^2/3

F(x) =

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11.

[-/1 Points]

DETAILS

OSCALC1 4.10.484.

MY NOTES

ASK YOUR TEACHER

Find the antiderivativeF(x)

of the functionf(x).

(UseCfor the constant of the antiderivative.)

f(x) = 0

F(x) =

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12.

[-/1 Points]

DETAILS

OSCALC1 4.10.490-498A.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the indefinite integral. (UseCfor the constant of integration.)

x6+

x

2dx

F(x) =

TutorialAdditional Materials

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13.

[-/1 Points]

DETAILS

OSCALC1 4.10.497.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Evaluate the integral. (UseCfor the constant of integration.)

18x³+2x+1

x³

dx

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14.

[-/1 Points]

DETAILS

OSCALC1 4.10.499-503.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Solve the initial value problem.

dy

dx

=5x³,y(0) =8

y=

TutorialAdditional Materials

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15.

[-/1 Points]

DETAILS

OSCALC1 4.10.499.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Solve the initial value problem.

f'(x) =x³,f(1) =7

f(x) =

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16.

[-/1 Points]

DETAILS

OSCALC1 4.10.503.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Solve the initial value problem.

f'(x) =4

x²

x²

4

,f(1) = 0

f(x) =

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17.

[-/2 Points]

DETAILS

OSCALC1 4.10.902.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

First findf'(x)

and then findf(x).

f''(x) =x³4x+8,f'(1) = 0,f(1) = 4

f'(x)=f(x)=

Tutorial

Analyze the graph off',

then list all intervals wherefis increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

increasingdecreasing

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2.

[-/7 Points]

DETAILS

OSCALC1 4.5.228.

MY NOTES

ASK YOUR TEACHER

Consider the function. (If an answer does not exist, enter DNE.)

f(x) =x+x²x³

(a)

Determine intervals wherefis increasing or decreasing. (Enter your answers using interval notation.)

increasingdecreasing

(b)

Determine the local minima and maxima off. (Enter your answers as comma-separated lists.)

locations of local minimax=locations of local maximax=

(c)

Determine intervals wherefis concave up or concave down. (Enter your answers using interval notation.)

concave upconcave down

(d)

Determine the locations of inflection points off. (Enter your answers as a comma-separated list.)

x=

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3.

[-/2 Points]

DETAILS

OSCALC1 4.6.280.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the horizontal and vertical asymptotes. (Letnrepresent an arbitrary integer. Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.)

f(x) =6x+1

x1

horizontaly=verticalx=

TutorialAdditional Materials

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4.

[-/3 Points]

DETAILS

OSCALC1 4.6.902.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Write all horizontal and vertical asymptotes for the function; list any removable discontinuities (holes). (If an answer does not exist, enter DNE.)

f(x) =x²2x

x²4

vertical asymptotex=horizontal asymptotey=holex=TutorialAdditional Materials

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5.

[-/3 Points]

DETAILS

OSCALC1 4.7.316.WA.TUT.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

A rectangular box which is open at the top can be made from an12-by-30-inch piece of metal by cutting a square from each corner and bending up the sides. Find the dimensions of the box with greatest volume, whereh= height,l= length,andw= width.(Note:let the width be determined by the12-inch side and the length by the30-inch side.)h=inl=inw=inTutorialAdditional Materials

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6.

[-/2 Points]

DETAILS

OSCALC1 4.7.345.

MY NOTES

ASK YOUR TEACHER

PRACTICE ANOTHER

Draw the given optimization problem and solve.

Find the dimensions (in units) of the closed cylinder with volumeV=54

units³that has the least amount of surface area.

radiusunitsheightunits

1..Find the change in volumedV(in units³) if the sides of a cube change from9to9.1.

dV=

units³

^5.A woman wishes to rent a house within 9 miles of her work. If she livesxmiles from her work, her transportation cost will becxdollars per year, while her rent will be16c

x+ 1

dollars per year,

wherecis a constant taking various situational factors into account. How far should she live from work to minimize her combined expenses for rent and transportation? Use the methods outlined in this section to find the minimum.

x=mi

6.Determine over what interval(s) (if any) the Mean Value Theorem applies. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

y= ln(7x9)

7.Determine the point(s), if any, at which the function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(t) =t+6

t²+7t+6

jump discontinuitiest=

n

removable discontinuitiest=infinite discontinuitiest=

other discontinuitiest=