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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

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
please answer
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]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
fg[[(45)]]²+45⅓x²=100
albert Reply
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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