Question

let f be a function such that f''(x)=6x+12

Answer 1

find f(X) if the graph of f is tangent to the line 4x-y=5 at the point (0,-5)

Answer 2

@phi

Answer 3

Did you integrate it ?

Answer 4

no

Answer 5

should i integrate 2 times?

Answer 6

Yes.

Answer 7

First step is find f'(x) by integrating once
remember to include the constant

Answer 8

3x^2+12x+c

Answer 9

2nd step is figure out the constant using the info given

Answer 10

-5=c

Answer 11

take the antiderivative of the equation
f'(x) = 3x^2 + 12x + c

the tangent line given is 4x - y = 5, or y = 4x - 5
so f'(x) = 4 when x = 0
use this information to solve for c

4 = 3(0) + 12(0) + c
c = 4

plug in c to get
f'(x) = 3x^2 +12x + 4

now take the antiderivative of this equation to get
f(x) = x^3 + 6x^2 + 4x + c

plug in the point (0, -5) to solve for c
-5 = 0 + 0 + 0 + c
c= -5

f(x) = x^3 + 6x^2 + 4x - 5

Answer 12

-5=c NO

the first derivative is the slope. you have to match the slope of the given line

Answer 13

4x-y=5 ?

Answer 14

yes and in slope-intercept form y= 4x-5 slope is 4

Answer 15

okay

Answer 16

what to do next

Answer 17

3x^2 +12x +c = slope
slope=4 at point (0,-5)

Answer 18

so c=4

Answer 19

yes. now you can integrate again to find f(x)
remember the constant

Answer 20

\[x^3+6x^2+4x+C\]

Answer 21

now how to find c

Answer 22

yes but you know a point on the curve (0, -5)

Answer 23

c=-5

Answer 24

yes, see maya's post for the computation in one write-up

Answer 25

thanks how to find average value of f(x) on the closed interval (-1,1)

Answer 26

find the area under the curve f(x)

the average f(x) is the height that gives the same area
height*width = area under the curve

so integrate between -1 and 1 and then divide by the width ( 2)

Answer 27

Does that make sense?

Answer 28

hmm but is their aother instead of graphing it

Answer 29

https://slader.s3.amazonaws.com/11/01/09/ca300f8f45970f319ed220978aadc955/0781b4a34c58a262b88a243f032b5f64/lateximg.png?Signature=tOScg1SekxIyLUCw0rVcPKaypI0%3D&Expires=1355071098&AWSAccessKeyId=AKIAJ7CBLRRHH3US6Z5Q

Answer 30

graphing?

I would integrate and divide by 2

Answer 31

got u what we got before (f(x)) should be integrated on -1 to 1 and divide by 2

Answer 32

right?

Answer 33

yes

Answer 34

thx