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f ( x ) = e x 3 x 2 + sin x

F ( x ) = e x x 3 cos ( x ) + C

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f ( x ) = e x + 3 x x 2

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f ( x ) = x 1 + 4 sin ( 2 x )

F ( x ) = x 2 2 x 2 cos ( 2 x ) + C

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For the following exercises, find the antiderivative F ( x ) of each function f ( x ) .

f ( x ) = x + 12 x 2

F ( x ) = 1 2 x 2 + 4 x 3 + C

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f ( x ) = ( x ) 3

F ( x ) = 2 5 ( x ) 5 + C

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f ( x ) = x 1 / 3 + ( 2 x ) 1 / 3

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f ( x ) = x 1 / 3 x 2 / 3

F ( x ) = 3 2 x 2 / 3 + C

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

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f ( x ) = sec 2 ( x ) + 1

F ( x ) = x + tan ( x ) + C

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f ( x ) = sin 2 ( x ) cos ( x )

F ( x ) = 1 3 sin 3 ( x ) + C

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f ( x ) = 1 2 csc 2 ( x ) + 1 x 2

F ( x ) = 1 2 cot ( x ) 1 x + C

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f ( x ) = csc x cot x + 3 x

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f ( x ) = 4 csc x cot x sec x tan x

F ( x ) = sec x 4 csc x + C

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f ( x ) = 8 sec x ( sec x 4 tan x )

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f ( x ) = 1 2 e −4 x + sin x

F ( x ) = 1 8 e −4 x cos x + C

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For the following exercises, evaluate the integral.

sin x d x

cos x + C

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3 x 2 + 2 x 2 d x

3 x 2 x + C

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( sec x tan x + 4 x ) d x

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( 4 x + x 4 ) d x

8 3 x 3 / 2 + 4 5 x 5 / 4 + C

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( x −1 / 3 x 2 / 3 ) d x

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14 x 3 + 2 x + 1 x 3 d x

14 x 2 x 1 2 x 2 + C

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( e x + e x ) d x

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For the following exercises, solve the initial value problem.

f ( x ) = x −3 , f ( 1 ) = 1

f ( x ) = 1 2 x 2 + 3 2

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f ( x ) = x + x 2 , f ( 0 ) = 2

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f ( x ) = cos x + sec 2 ( x ) , f ( π 4 ) = 2 + 2 2

f ( x ) = sin x + tan x + 1

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f ( x ) = x 3 8 x 2 + 16 x + 1 , f ( 0 ) = 0

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f ( x ) = 2 x 2 x 2 2 , f ( 1 ) = 0

f ( x ) = 1 6 x 3 2 x + 13 6

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For the following exercises, find two possible functions f given the second- or third-order derivatives.

f ( x ) = e x

Answers may vary; one possible answer is f ( x ) = e x

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f ( x ) = cos x

Answers may vary; one possible answer is f ( x ) = sin x

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f ( x ) = 8 e −2 x sin x

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A car is being driven at a rate of 40 mph when the brakes are applied. The car decelerates at a constant rate of 10 ft/sec 2 . How long before the car stops?

5.867 sec

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In the preceding problem, calculate how far the car travels in the time it takes to stop.

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You are merging onto the freeway, accelerating at a constant rate of 12 ft/sec 2 . How long does it take you to reach merging speed at 60 mph?

7.333 sec

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Based on the previous problem, how far does the car travel to reach merging speed?

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A car company wants to ensure its newest model can stop in 8 sec when traveling at 75 mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

13.75 ft/sec 2

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A car company wants to ensure its newest model can stop in less than 450 ft when traveling at 60 mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

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For the following exercises, find the antiderivative of the function, assuming F ( 0 ) = 0 .

[T] f ( x ) = x 2 + 2

F ( x ) = 1 3 x 3 + 2 x

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[T] f ( x ) = sin x + 2 x

F ( x ) = x 2 cos x + 1

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[T] f ( x ) = 1 ( x + 1 ) 2

F ( x ) = 1 ( x + 1 ) + 1

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[T] f ( x ) = e −2 x + 3 x 2

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For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.

If f ( x ) is the antiderivative of v ( x ) , then 2 f ( x ) is the antiderivative of 2 v ( x ) .

True

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If f ( x ) is the antiderivative of v ( x ) , then f ( 2 x ) is the antiderivative of v ( 2 x ) .

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If f ( x ) is the antiderivative of v ( x ) , then f ( x ) + 1 is the antiderivative of v ( x ) + 1 .

False

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If f ( x ) is the antiderivative of v ( x ) , then ( f ( x ) ) 2 is the antiderivative of ( v ( x ) ) 2 .

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Chapter review exercises

True or False ? Justify your answer with a proof or a counterexample. Assume that f ( x ) is continuous and differentiable unless stated otherwise.

If f ( −1 ) = −6 and f ( 1 ) = 2 , then there exists at least one point x [ −1 , 1 ] such that f ( x ) = 4 .

True, by Mean Value Theorem

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If f ( c ) = 0 , there is a maximum or minimum at x = c .

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Questions & Answers

Find the derivative of g(x)=−3.
Abdullah Reply
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Abish
How can I help you?
Tlou
evaluate the following computation (x³-8/x-2)
Murtala Reply
teach me how to solve the first law of calculus.
Uncle Reply
what is differentiation
Ibrahim Reply
f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
Omar Reply
what is absolute value means?
Geo Reply
Chicken nuggets
Hugh
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MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
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Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
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Practice Key Terms 3

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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