onegirl 2 years ago Find an equation of the tangent line y = f(x) at x = a. f(x) = sin 4x, a = pi/8

1. zepdrix

Hmm I think you've done a few of these types of problems. So I'm just curious, which result makes more sense to you. Seeing our tangent line in slope-intercept form: $$\large y=mx+b$$, Or in point-slope form: $$\large y-y_o=m(x-x_o)$$.

2. onegirl

in slope intercept from

3. zepdrix

Ok we will have 2 pieces we need to find. $$\large m$$ is the slope of our tangent line. Take the derivative of your function, and evaluate it at $$\large x=\pi/8$$. That will be your $$\large m$$ value.

4. onegirl

okay hold on

5. onegirl

when i find the derivative a = pi/8 is not int he equation right?

6. onegirl

the*

7. onegirl

i got sin(4)

8. zepdrix

Take the derivative of sin(4x) first. Don't plug a=pi/8 in until after you've taken a derivative.

9. onegirl

okay

10. onegirl

i got 4 cos(4x)

11. zepdrix

ok good, now plug in x=pi/8 :D

12. onegirl

ok

13. onegirl

i got 0 after i plugged in and solved

14. zepdrix

Ok sounds good. So we've determined that the slope $$\large m$$ of our tangent line will be 0. $\large y=mx+b \qquad \rightarrow \qquad y=0x+b$ Now we just need to find the $$\large b$$ value.

15. zepdrix

To find it, we'll need to plug in a coordinate pair that falls on the line. Well, since our line is tangent to the curve at x=pi/8, we can plug x=pi/8 into that function to get a corresponding y value. This will be the coordinate pair we'll plug into our tangent line.

16. onegirl

ok

17. onegirl

so i should plug in pi/8 into y = mx + b?

18. zepdrix

no, plug it into your original function, before you took the derivative of it.

19. onegirl

ohh okay

20. onegirl

so sin 4(pi/8) ?

21. onegirl

okay so i got 1

22. zepdrix

So that gives us a coordinate pair of $$\large \left(\dfrac{\pi}{8},\;1\right)$$. Plug that into your tangent function to solve for $$\large b$$.

23. onegirl

ok so y = 0x + b so i should plug it in place of b?

24. zepdrix

No, your coordinate pair is $$\large (x,\;y)$$. Plug it into those.

25. onegirl

ohh ok got it

26. onegirl

okay so 1 = 0(pi/8) + b

27. zepdrix

So what is your b value? :3

28. onegirl

Lol sorry i got 1 :)

29. onegirl

b = 1

30. zepdrix

Ok sounds good. So if we plug that missing $$\large b$$ into our tangent line equation, we get,$\large y=0x+1$Which can be simplified to,$\large y=1$

31. onegirl

okay

32. zepdrix

I guess it's really 3 steps. We had to find ~$$\large m$$ ~$$\large b$$ ~A Coordinate Pair.

33. zepdrix

It's not too difficult, just a lot of silly letters :) Keep practicing!

34. onegirl

okay so our equation is complete? and yes just lot to do and thanks

35. zepdrix