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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 integral of tan
Gagan Reply
Differentiate each from the first principle. y=x,y=1/x
Abubakar Reply
I need help with calculus. Anyone help me.
Macquitasha Reply
yes
Pradip
formula for radius of curvature
mpradeepa Reply
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Usman
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Macquitasha
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Sumit
integration seems interesting
Fund Reply
it's like a multiple oparation in just one.
Efrain
Definitely integration
ROHIT Reply
tangent line at a point/range on a function f(x) making f'(x)
Luis
Principles of definite integration?
ROHIT
For tangent they'll usually give an x='s value. In that case, solve for y, keep the ordered pair. then find f(x) prime. plug the given x value into the prime and the solution is the slope of the tangent line. Plug the ordered pair into the derived function in y=mx+b format as x and y to solve for B
Anastasia
parcing an area trough a function f(x)
Efrain
Find the length of the arc y = x^2 over 3 when x = 0 and x = 2.
Jade Reply
integrate x ln dx from 1 to e
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application of function
azam Reply
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azam Reply
what ?
Bunyim
defination of math
azam
application of function
azam
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azam
what is a circle
Ronnie
math is the science, logic, shape and arrangement
Boadi
a circle is a hole shape
Jianna
a whole circumference have equal distance from one point
azam
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azam
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Agboke
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Bruce
show that the f^n f(x)=|x-1| is not differentiable at x=1.
Mohit Reply
is there any solution manual to calculuse 1 for Gilbert Strang ?
Eng Reply
I am beginner
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Badar
just began, bois!!
Luis
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Jianna
what is mathematics
Henry Reply
logical usage of numbers
Leo
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David
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Bruce
mathematics seems to be anthropocentric deductive reasoning and a little high order logic. I only say this because I can only find two things going on which is infinitely smaller than 0 and anything over 1
David
More comprehensive list here: ***onetonline.org/find/descriptor/result/1.A.1.c.1?a=1
Bruce
so how can we differentiate inductive reasoning and deductive reasoning
Henry
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is there anyone who can guide me in learning the mathematics easily
Sabir
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Leo
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in calculas,does a self inverse function exist
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Moises
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ashwini
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Agboke
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Agboke
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Henry
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Leo
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Adewale
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wasim
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Agboke
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Leo
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Luis
dy/dx= 1-cos4x/sin4x
Alma Reply
what is the derivatives of 1-cos4x/sin4x
Alma
what is the derivate of Sec2x
Johar
d/dx(sec(2 x)) = 2 tan(2 x) sec(2 x)
AYAN
who knows more about mathematical induction?
Agboke
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matbakh
Yes
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Gagan
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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