Christos
  • Christos
calculus Help me solve #44 please https://www.dropbox.com/s/r8mq9jvwwaudsut/Screenshot%202015-12-07%2014.14.19.png?dl=0
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
For first one we have x= pi and y= xcos(3x) have u done any work so far ?
anonymous
  • anonymous
okay first put x=pi in y= x cos(3x) so y = ?
Christos
  • Christos
I have solved that one, can you help me with #44 please

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anonymous
  • anonymous
oh wait a bit ..:)
zepdrix
  • zepdrix
\[\large\rm y=\sin(1+x^3)\]So for your derivative,\[\large\rm y'=\cos(1+x^3)\cdot(1+x^3)'\]chain rule, ya?
Christos
  • Christos
yes
zepdrix
  • zepdrix
\[\large\rm y'=\cos(1+x^3)\cdot(3x^2)\]The derivative evaluated at x=-3 will give us the `slope` of our tangent line,\[\large\rm Y=mx+b\] \[\large\rm y'(x)=3x^2 \cos(1+x^3)\]\[\large\rm y'(-3)=m\] Plug it in. Figure out your slope. Do it. :)
Christos
  • Christos
whats cos(8)
Christos
  • Christos
@zepdrix
Christos
  • Christos
cos(-8)
zepdrix
  • zepdrix
(-3)^3 is not anywhere near 8 silly >.<
zepdrix
  • zepdrix
\(\rm (-3)^3\ne-9\)
Christos
  • Christos
cos(-26) ? :D
zepdrix
  • zepdrix
Ok good. And it has a coefficient in front of it, ya? Like a -9 or something. Or 27, ya.
Christos
  • Christos
yes
zepdrix
  • zepdrix
\[\large\rm m=27\cos(-26)\]
zepdrix
  • zepdrix
Oh we can make one nice simplification. Since cosine is an even function, it follows this rule:\[\large\rm \cos(-x)=\cos(x)\]
zepdrix
  • zepdrix
Therefore,\[\large\rm m=27\cos(-26)=27\cos(26)\]
Christos
  • Christos
ok
zepdrix
  • zepdrix
So for our line that we're trying to construct, let's plug in what we have so far,\[\large\rm Y=\color{royalblue}{m}x+b\]\[\large\rm Y=\color{royalblue}{27\cos(26)}x+b\]
Christos
  • Christos
should we evaluate cos(26)
zepdrix
  • zepdrix
I don't know. Are you submitting this online? Or just textbook stuff? I think it looks a lot nicer if we leave it in exact form like this. But if your instructions want decimal approximation we can do that.
Christos
  • Christos
its textbook stuff
Christos
  • Christos
I just wanna know how to do it
zepdrix
  • zepdrix
There is no fancy way "to do it" unfortunately :) The angle measure of 26 radians doesn't correspond to any "special measure" like pi/4 would for example. So you either: Leave it as is, or use a calculator ^^
Christos
  • Christos
i see
zepdrix
  • zepdrix
\[\large\rm Y=27\cos(26)x+b\]To figure out this missing b value, we need to be able to plug in a coordinate point, then we can solve for b.
zepdrix
  • zepdrix
But what point can we use? Hmm... AH! Well recall that the tangent line "touches" the curve at this particular point! So the tangent line Y and the curve y both share the coordinate point (-3, ? )
Christos
  • Christos
how do you know its radians and not degrees ?
zepdrix
  • zepdrix
Ah good question :) Always assume it's radians unless you see the little degree bubble. That's, ya
Christos
  • Christos
(-3,27*cos(26)*-3+b)
Christos
  • Christos
so pi is radians right?
zepdrix
  • zepdrix
No no. To find our y-coordinate, we need to plug -3 back into the `original function`.
zepdrix
  • zepdrix
Yes, pi is measured in radians :3
Christos
  • Christos
(-3,sin(-26))
Christos
  • Christos
I saw people refering to pi as in 180 degrees
Christos
  • Christos
so then cos(3.14) = cos(180) in degrees
Christos
  • Christos
right?
zepdrix
  • zepdrix
Ah, yes good :) When we're talking about the `unit circle`, a radial distance of pi is half of the full circle, which is also 180 degrees. So yes, that works out nicely.
Christos
  • Christos
ic
zepdrix
  • zepdrix
We'll apply the fact that sine is an odd function, so it follows this property:\[\large\rm \sin(-x)=-\sin(x)\]So that gives us our ordered pair: \[\large\rm \left(-3,-\sin(26)\right)\]
zepdrix
  • zepdrix
\[\large\rm Y=27\cos(26)x+b\]both the curve and the tangent line share this point. So now we'll plug this coordinate point into our tangent line, and solve for b.
Christos
  • Christos
thanks a lot for help!!
zepdrix
  • zepdrix
np

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