Question

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

Answer 1

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: \(\large y-y_o=m(x-x_o)\).

Answer 2

in slope intercept from

Answer 3

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.

Answer 4

okay hold on

Answer 5

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

Answer 6

the*

Answer 7

i got sin(4)

Answer 8

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

Answer 9

okay

Answer 10

i got 4 cos(4x)

Answer 11

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

Answer 12

ok

Answer 13

i got 0 after i plugged in and solved

Answer 14

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.

Answer 15

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.

Answer 16

ok

Answer 17

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

Answer 18

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

Answer 19

ohh okay

Answer 20

so sin 4(pi/8) ?

Answer 21

okay so i got 1

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

ohh ok got it

Answer 26

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

Answer 27

So what is your b value? :3

Answer 28

Lol sorry i got 1 :)

Answer 29

b = 1

Answer 30

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,\[\large y=1\]

Answer 31

okay

Answer 32

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

Answer 33

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

Answer 34

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

Answer 35

Yes, that's our final answer.