Find an equation of the tangent line to the graph f(x)=2-7x^2 at (2,-26)

Question

amistre64 · Answer

what are 2 things we need to know to define a line?

anonymous · Answer

two points?

amistre64 · Answer

well, 2 points are useful yes, but we only have one point given to us ... what else could we use to define a line with?  a point and ....

anonymous · Answer

slope?

amistre64 · Answer

correct, so a point and a slope can be used to define a line.  its called the point slope form of the equation.

now, how do we determine the slope of a curve at a given point?

anonymous · Answer

the change in y over the change in x? so the point slope equation?

amistre64 · Answer

hmm, im assuming you know what a derivative is for this to work.

anonymous · Answer

yeah but I don't really get derivatives

anonymous · Answer

I know there's 4 types

amistre64 · Answer

you should have a simple rule you can apply

constant rule, and a power rule

amistre64 · Answer

$$\frac{d}{dx}k=0$$
$$\frac{d}{dx}x^n=n~x^{n-1}$$

anonymous · Answer

I'm just having a hard time applying the equation to these rules

amistre64 · Answer

9 - x^2
^   ^
^    this is a power of x
^
and this is a constant

amistre64 · Answer

what is the derivative of 9?

what is the derivative of -x^2?

amistre64 · Answer

e3rr,  -7x^2  ... forgot the equation we were dealing with :/

anonymous · Answer

so for 2 it would be 0 and -7x^2 would be -7x^-6?

amistre64 · Answer

$$\frac{d}{dx}(2-7x^2)$$
$$\frac{d}{dx}(29)-\frac{d}{dx}(7x^2)$$
$$\frac{d}{dx}(2)-7\frac{d}{dx}(x^2)$$
$$0-7(2x^{2-1})$$

anonymous · Answer

how did you get 29?

amistre64 · Answer

typo, slip of the fingers.

anonymous · Answer

can you please explain how you got the answer step by step?

amistre64 · Answer

i jsut did ... except for 29 which is spose to be a 2 instead

use the constant rule for the constant, and the power rule for the power of x

there is also a 'pull out' rule which lets us pull out that 7 to the outside.

amistre64 · Answer

your -7x^2 derivative is off ... try it again

amistre64 · Answer

the format for the tangent line is just gonna be:

y - f(a) = f'(a) (x-a)

so as long as we can determine f'(a) we are set

anonymous · Answer

sorry my book shows different explanations so I'm a little confused right now

amistre64 · Answer

tell me your thoughts then, what does your book explain?

anonymous · Answer

it basically shows me all these rules but all these variables are confusing me

anonymous · Answer

but the examples only show me if I have a point and slope

amistre64 · Answer

what is your rule for powers of x?

amistre64 · Answer

can you take a picture of it?

anonymous · Answer

For any real number k if y=x^k then d/dx(x^h)= k*x^k-1

amistre64 · Answer

let k=2

x^2 become 2x^(2-1)

or simply 2x

amistre64 · Answer

so our derivative is

-7x^2  derives to -7(2x),  at x=2 that is ... -7(4)

anonymous · Answer

so the 2 becomes x?

amistre64 · Answer

out point (2,-26) defines the value of x as 2

amistre64 · Answer

*our point ...

amistre64 · Answer

2x, when x=2 is just ... 2(2)  right?

anonymous · Answer

how did x^2 become 2x?

amistre64 · Answer

by the derivative rule you posted.  let k=2

amistre64 · Answer

x^k derives to  k*x^(k-1)

anonymous · Answer

Ohhhh I get it now

amistre64 · Answer

by first principles of limit

$$\lim_{h\to0}\frac{(x+h)^2-x^2}{(x+h)-x}$$
$$\lim_{h\to0}\frac{x^2+h^2+2xh-x^2}{h}$$
$$\lim_{h\to0}\frac{h^2+2xh}{h}$$
$$\lim_{h\to0}\frac{h(h+2x)}{h}$$
$$\lim_{h\to0}(h+2x)=2x$$

anonymous · Answer

but I'm still pretty confused

amistre64 · Answer

the 'rules' are really a table of outcomes that can be applied so that you dont have to use the long limiting process each and every time.

anonymous · Answer

so now that we know the derivative of -7x^2 to be -7(2x) which is -7(2*2)= -28
what do we do next?

amistre64 · Answer

well, thats our slope of our tangent line.  the rest is from algebra class.

amistre64 · Answer

construct the line using your slope and the stated point

anonymous · Answer

y=mx+b

anonymous · Answer

or point slope equation
y-y1=m(x-x1)

amistre64 · Answer

y - f(a) = f'(a) (x-a)

or rewritten as

y = f'(a) (x-a) +f(a)

amistre64 · Answer

point slope equation

anonymous · Answer

y-(-26)=-28(x-2)

amistre64 · Answer

thatll do it

anonymous · Answer

y=-28x+30

anonymous · Answer

could I get a two out of it to be 2(-14x+15)? or is okay like that?

amistre64 · Answer

usually we dont factor it, keep it in y=mx+b form is conventional

anonymous · Answer

so that's my final answer? thank you! is there any way I reward you on here? it's my first time using this site

amistre64 · Answer

there should be a 'best response' button to everything ive posted.  clicking it gives out a medal.  but saying thank you and learning a little bit is reward enough for me to be honest.

anonymous · Answer

okay thank you!!

amistre64 · Answer

youre welcome :)  and good luck