Question

The line y=3x+7 is tangent to the graph y=f(x)at x=4. What is f(4)? What Is f′(4).

Answer 1

okay let's have some fun.

Answer 2

f(x) = 7x - x^2
f'(x) = 7- 2x
7- 2x=3
2x= 4
x=2

y = 7x - x^2
y = 14 - 4
y = 10

slope = 3, point (2, 10)
y - y1 = m(x - x1) >> slope point equation of the line.
y -10 = 3(x - 2)
y = 3x - 6+10
y= 3x + 4

Answer 3

looks like he got it

Answer 4

Please explain. Thanks.

Answer 5

which is f(4) and f'(4)?

Answer 6

and how did yo uget your first equation? antiderivative?

Answer 7

nvm

Answer 8

still an explanation would be nice

Answer 9

u need it?

Answer 10

i'll run you through again okay?

Answer 11

give me a medal :]

Answer 12

yes plz!

Answer 13

y=3x+7 is tangent to the graph y=f(x) @ x=4. What's f(4)? What's f'(4)?

Answer 14

y-f(x)=f'(x)(x-a)

Answer 15

a=4, f(x)=unknown, f'(x)=unknown

Answer 16

tangent of graph is found by the formula above

Answer 17

3x+7=f'(x)(x-a)+f(x)

Answer 18

tangents have m=-1xrecipricol

Answer 19

so f(x)=-1/3x+c

Answer 20

plug in point x=4

Answer 21

-1/3(4)+c=f(x)

Answer 22

c being 7?

Answer 23

or what is c?

Answer 24

hmm let me do this on paper, i'll refine it

Answer 25

ahh...you need to integrate

Answer 26

easy

Answer 27

dirty tricksters as I thought.... then f'(4) is just 19 right?

Answer 28

y=19

Answer 29

it's tricky, i'll show you

Answer 30

wups did you mean integrate the function you found?

Answer 31

ahh kk

Answer 32

@ x=4, y=19

Answer 33

so you plug that into the integral of tangent

Answer 34

and find intercept (c)

Answer 35

c= -33

Answer 36

how is it -33?

Answer 37

henceforth, \[f(x)=3/2x^2+7x-33\]

Answer 38

at the point x=4, y=19 right?

Answer 39

the tangent is essentially the line of the gradient m or dy/dx

Answer 40

at that given point (4,19)

Answer 41

so the line is the integral of the tangent at that point true?

Answer 42

wait..

Answer 43

it's messy when you have too many unknowns

Answer 44

yes i was correct

Answer 45

okay all the answers:
\[f(x)=3/2x^2+7x-33\]
f(4)=639
f'(4)=19

Answer 46

i overcomplicated things a lot sorry