Answer:
a. y=4
b. graph the given and y=4
Explanation:
The given is y= 8 sin(x) cos(x)
to get y' which is the slope of the tangent line we take the derivative
for this problem, the derivative involves the constant rule (which states that the derivative a constant times a value is just the constant times the derivative of said value) and the multiplication rule (assuming that the two parts in question are represented by the variables a and b then the derivative of a*b = a*b'+b*a')
thus we get 8*(sin(x)*-sin(x)+cos(x)*cos(x))
-8sin^2(x)+8cos^2(x)
You can also leave the 8 differentiated out depending on the form you need.
Now that we have the derivative we can plug pi/4 into this equation to get the slope of the tangent line at that point
=8sin^2(pi/4)+8cos^2(pi/4)
this would equal -8*1/2 +8*1/2 which equals 0
the slope of the tangent at this point would be zero
To get the point for point slope form we plug the pi/4 into the given equation to get a y value and get the point (pi/4, 4)
this means the point slope equation would be
y-4=0(x-pi/4)
y-4=0
y=4
To graph you would just enter the given equation and the equation for the tangent line into a graphing software or calculator and see if the line is tangent at the point (pi/4, 4)