Given `f(x)=3sinxcosx` :
(1) Find the equation of the line tangent to the function when `x=pi` .
The slope of the tangent line is the first derivative evaluated at `x=pi`
`f'(x)=3cos^2x-3sin^2x=3cos2x` (using the extended product property and a trig identity)
`f'(pi)=3cos(2pi)=3` so the slope is 3. The tangent passes through the point `(x,f(x))=(pi,f(pi))` where `f(pi)=3sinpicospi=0` .
So with slope 3 and the point `(pi,0)` , the equation of the line is:
(2) When does the rate of change equal -3/2?
Set the first derivative equal to -3/2:
Then `2x=(2pi)/3 + 2kpi` or `2x=(4pi)/3+2kpi` where `kinZZ` .
Thus `x=pi/3+kpi` or `(2pi)/3+kpi`