a) Find an equation of a line that is tangent to the curve f(x)=2cos3x and whose slope is a maximum.
b) Is this the only possible solution? Explain. If there are other possible solutions, how many solutions are there?
For a.. I know how to find the tangent of a curve at x=something.. but this question asks to find the tangent whose slope is a maximum; what would I have to do to determine that?
to find when the derivative is at a maximum, we take the derivative once more
and we find that it's at a maximum when .
So than we evaluate the first derivative to see what the slope is at .
so the slope, which happens to be the optimal slope, is -6.
To find where on the graph we should draw the tangent line, we evaluate using the regular function and we find that
using this info we can do some algebra to get the equation of this line which is
b) This isn't the only possible solution. Remember when we took the second derivative
and set it equal to 0 and got . Then we evaluated using the original function, , and we got .
Essentially what this is telling us is that the original function hits a slope of when ever it touches the x-axis (equal to 0).
Since it's a trig function, it continues to hit 0 to infinity.
The real answer (when the function hits a maximum slope) is where is an integer.
uh-oh.. i just notcied a mistake. in my original post i said the derivative is
it's supposed to be
this means that the second derivative is also wrong. it was
it's supposed to be
ok now we're back on track. We set the second derivative equal to zero, and we find the slope is at a max when
Everything's correct from there.. i mention the second derivative again; replace it with the corrected one...
then we divide both sides by -18
then we take the inverse cos function of both sides (sometimes it's written with a power of minus one but i prefer . so..
.. inverse cos of 0 is 90 degrees (or pi/6 radians)
has a period of , and given the general shape of the cosine curve, the maximum slope will occur when ... .
this maximum slope will occur at all values of , and will be equal to 6.