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

1. amistre64

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

2. anonymous

two points?

3. amistre64

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 ....

4. anonymous

slope?

5. amistre64

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?

6. anonymous

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

7. amistre64

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

8. anonymous

yeah but I don't really get derivatives

9. anonymous

I know there's 4 types

10. amistre64

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

11. amistre64

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

12. anonymous

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

13. amistre64

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

14. amistre64

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

15. amistre64

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

16. anonymous

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

17. amistre64

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

18. anonymous

how did you get 29?

19. amistre64

typo, slip of the fingers.

20. anonymous

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

21. amistre64

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.

22. amistre64

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

23. amistre64

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

24. anonymous

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

25. amistre64

26. anonymous

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

27. anonymous

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

28. amistre64

what is your rule for powers of x?

29. amistre64

can you take a picture of it?

30. anonymous

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

31. amistre64

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

32. amistre64

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

33. anonymous

so the 2 becomes x?

34. amistre64

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

35. amistre64

*our point ...

36. amistre64

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

37. anonymous

how did x^2 become 2x?

38. amistre64

by the derivative rule you posted. let k=2

39. amistre64

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

40. anonymous

Ohhhh I get it now

41. amistre64

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

42. anonymous

but I'm still pretty confused

43. amistre64

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.

44. anonymous

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?

45. amistre64

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

46. amistre64

construct the line using your slope and the stated point

47. anonymous

y=mx+b

48. anonymous

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

49. amistre64

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

50. amistre64

point slope equation

51. anonymous

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

52. amistre64

thatll do it

53. anonymous

y=-28x+30

54. anonymous

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

55. amistre64

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

56. anonymous

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

57. amistre64

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.

58. anonymous

okay thank you!!

59. amistre64

youre welcome :) and good luck