thuyvy 3 years ago Find equations of all tangents to the curve f(x)=9/x that have slope -1

1. ivanmlerner

At first you have to take the derivative of f. Do you understand why is that?

2. thuyvy

yes I do, I got -9/x2

3. ivanmlerner

Ok, now what is the meaning of the derivative? Knowing this meaning can you guess what the next step would be?

4. thuyvy

that is when a function change as the input change, but I'm stuck at the 2nd step what shall I do next? plug in -1 slope as x?

5. ivanmlerner

Yes, in other words it is the rate of change of a function. But the meaning I was thinking of is the same, but a little diferent. When the input increases in a really small amount, the value changes too. Since those two changes are very small, the rate of change is approximately the change in the value divided by the increase in the input. Do you understand what I just said?

6. thuyvy

not really, can you lead me into solving this? what is the second step I should take?

7. ivanmlerner

You need to understand it before gettingg to the next step. But if you want to know, the derivative of a function at x is the slope of its tangent line at this point, that was what I was trying to show you. Then, you should put -9/x2=-1, that is the slope, to find in which points the line passes.

8. thuyvy

Yes I got to this step, when I was trying to solve for x I got +- 3

9. thuyvy

but the answer that it should be is y=-x-6 and y = -x+6

10. thuyvy

so I must have done something wrong right?

11. ivanmlerner

Wait, you just found out one point in wich the line passes, it is not the answer yet

12. ivanmlerner

What you did so far is correct

13. thuyvy

oh ic okay, so the other point is 3 since 9/x = y and x is 3 therefore y = 3 right?

14. ivanmlerner

or -3, but yes, thats correct

15. thuyvy

so now what's next?

16. ivanmlerner

Now, you know a point that belongs to the line, and its slope. The general formula for a line is y=ax+b, and you alrady have a, now put the point you know that belongs to it, and you will find b.

17. FaisalAyaz

use slope intercept form now

18. thuyvy

y = mx+b --> -3=-1x+b

19. thuyvy

sorry, I'm kinda slow at this point, how do I set this = to 9?

20. ivanmlerner

You dont need to, you know x and y, from the points (-3, -3) and (3, 3) that this tangent passes through, and you know a. Now: -3=-1*(-3)+b1 3=-1*3+b2

21. thuyvy

oh okay

22. thuyvy

I got it from here

23. ivanmlerner

Before, you found out in which points the tangent have this slope, then, you found out the value of the function at this point, and now you use those information to get the formulas for the tangent lines. You seem a little bit confused with what the numbers you are getting mean.

24. thuyvy

I got it thanks so much for being so patience :)

25. ivanmlerner

26. thuyvy

can I ask you something else?

27. ivanmlerner

Of course

28. thuyvy

same type of problem but it's f(x) = square root of (x+9)

29. thuyvy

slope 1/2 I did the derivatives and solve for x I got x= -1

30. thuyvy

I plugged x= -1 back to original equation I got y = +/- 2 sqrt 2 then I use slope intercept to find equations, I got y= 1/4+9/4 sqrt 2

31. thuyvy

but the answer should be y=1/4+13/4 where did I do wrong?

32. ivanmlerner

You got the x wrong.

33. ivanmlerner

Or the derivative

34. thuyvy

original fx is sqrt(x=9) derivative is 1/2(x+9)^1/2

35. thuyvy

Where did I do wrong.

36. ivanmlerner

its (1/2)(x+9)^(-1/2)

37. thuyvy

Yes i got that.. But solving for x=-1

38. ivanmlerner

Well, thats wrong, did you see the minus, in the exponent?

39. thuyvy

No i said that mine is 1/{2(x+9)^1/2 same thing

40. ivanmlerner

$\frac{ 1 }{ 2 }=\frac{ 1 }{ 2\sqrt{x+9} }\rightarrow \sqrt{x+9}=1\rightarrow x+9=\pm1\rightarrow x=-10 or -8$

41. thuyvy

the slope is 1/4 not 1/2

42. ivanmlerner

sorry, didnt ssee you posting. Then the left side of the equation will be 1/4, and sqrt(x+9)=2 and x+9=+-4, wich gives us: x=-5, and x=-13

43. ivanmlerner

Can you go from there?

44. thuyvy

let me try to do calculation, I did the calculation but got +/- 2 sqrt 2.. give me 1 minute please

45. ivanmlerner

Sure

46. thuyvy

I got it, I think I got messed up when I was trying to square to get rid of sq rt. Thanks Do you have time to help me on other problems?

47. ivanmlerner

I'm almost leaving actually, but I'm here almost every day.

48. thuyvy

quick question on solving this problem $z(4z+7) - x(4x+7) / (z-x)$

49. thuyvy

I can combine (z-x) then cancel top and bottom right?

50. thuyvy

so what is have left is (4z+7) * (4x+7) right?

51. ivanmlerner

No, be carefull, think of what is being divided, when you have something that is not being divided you cannot sum them withou a common denominator

52. thuyvy

oh maybe that's why. thanks