anonymous
  • anonymous
Find an equation of the tangent line y = f(x) at x = a. f(x) = sin 4x, a = pi/8
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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zepdrix
  • 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: \(\large y-y_o=m(x-x_o)\).
anonymous
  • anonymous
in slope intercept from
zepdrix
  • 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.

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anonymous
  • anonymous
okay hold on
anonymous
  • anonymous
when i find the derivative a = pi/8 is not int he equation right?
anonymous
  • anonymous
the*
anonymous
  • anonymous
i got sin(4)
zepdrix
  • zepdrix
Take the derivative of sin(4x) first. Don't plug a=pi/8 in until after you've taken a derivative.
anonymous
  • anonymous
okay
anonymous
  • anonymous
i got 4 cos(4x)
zepdrix
  • zepdrix
ok good, now plug in x=pi/8 :D
anonymous
  • anonymous
ok
anonymous
  • anonymous
i got 0 after i plugged in and solved
zepdrix
  • 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.
zepdrix
  • 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.
anonymous
  • anonymous
ok
anonymous
  • anonymous
so i should plug in pi/8 into y = mx + b?
zepdrix
  • zepdrix
no, plug it into your original function, before you took the derivative of it.
anonymous
  • anonymous
ohh okay
anonymous
  • anonymous
so sin 4(pi/8) ?
anonymous
  • anonymous
okay so i got 1
zepdrix
  • 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 \(\large b\).
anonymous
  • anonymous
ok so y = 0x + b so i should plug it in place of b?
zepdrix
  • zepdrix
No, your coordinate pair is \(\large (x,\;y)\). Plug it into those.
anonymous
  • anonymous
ohh ok got it
anonymous
  • anonymous
okay so 1 = 0(pi/8) + b
zepdrix
  • zepdrix
So what is your b value? :3
anonymous
  • anonymous
Lol sorry i got 1 :)
anonymous
  • anonymous
b = 1
zepdrix
  • 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,\[\large y=1\]
anonymous
  • anonymous
okay
zepdrix
  • zepdrix
I guess it's really 3 steps. We had to find ~\(\large m\) ~\(\large b\) ~A Coordinate Pair.
zepdrix
  • zepdrix
It's not too difficult, just a lot of silly letters :) Keep practicing!
anonymous
  • anonymous
okay so our equation is complete? and yes just lot to do and thanks
zepdrix
  • zepdrix
Yes, that's our final answer.

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