Question

f(4)=1
f ' (x)=-2
Find an equation of the tangent line to f(x)at the point x=4.

Answer 1

to represent a line by algebraic forms, you need to know two things. slope and the point it passes.
those informations are in the f(4)=1 f'(x)=-2

Answer 2

I have to find the slope and y intercept so I don't have any of that either

Answer 3

I have no idea where to start I really need help please anyone

Answer 4

y-f(4)=f'(x)(x-4)
y-1=-2(x-4)
2 x+y-9=0

Answer 5

so then the 2x+y=0 is the tangent equation

Answer 6

oh sorry I mean 2x+y-9=0 is the tangent equation

Answer 7

is this correct

Answer 8

you understand that f'(a) is a slope of the tangent line at x=a right?

Answer 9

and f'(x)=-2 says f'(4)=-2

Answer 10

also, y intercept is just one of the points that line passes.
you can use other points that line pass.
f(4)=1 says the tangent line passes (4,1)

Answer 11

so assuming that you understand y=m(x-a)+b,
y=(-2)(x-4)+1

Answer 12

or in other form, 2x+y=9

Answer 13

ok so then are yousaying that the slope is -2 and the y intercept is 1

Answer 14

defintion of y intercept is the value of y at x=0

Answer 15

y intercept is of course 9..

Answer 16

ok yup just got that so then the slope is -2

Answer 17

yeah

Answer 18

ok got that then what it I have t estimate f(7) then what so I do

Answer 19

i don't understand what you want to know.
what is t?

Answer 20

sorry bad typing I meant how do I find the estimate of f(7)

Answer 21

oh
so estimation is something close to the real solution right?

Answer 22

yes

Answer 23

in this case, (assuming you don't know integral calculus) you can't figure out the exact f(7) because you don't know the rule

Answer 24

right it just has to be an estimate

Answer 25

yeah. well you can use your tangent line to make that approximation

Answer 26

ok so how do you do that

Answer 27

you plug 7 into x in your equation of tangent line

Answer 28

actually the whole point of calculus is that if you cut curves very small, it estimates to straight lines

Answer 29

so then is the esimate of f(7)=-5

Answer 30

oh ok

Answer 31

it is called 'linear approximation'
and your answer is right

Answer 32

yay ok thanks for all your help