Raymond B. answered • 07/13/19
Math, microeconomics or criminal justice
Instantaneous rate of change is the first derivative or the slope of the tangent of the function at x= pi/3
first derivative is 2cos(2x+4) At x=pi/3 2cos(2pi/3+4)=1.96447
2pi/3 + 4 = 6.09440
cos(6.0944)=0.982223
2(.98223)=1.96447
Alternatively use cos(x+y)=cosxcosy-sinxsiny with x=2pi/3 and y=4 and calculate
cos(2pi/3)=-0.5, sin2pi/3=.86603
cos4=-0.65364, sin4=-0.75680 and multiply the sines, multiply the cosines, sum them, multiply by 2
Answer is 1.96447 or almost 2
IF you work it two ways and get the same answer that is good evidence there's no mistake.
Another approach is estimate the answer. Look at the sine function. It's sine of 2pi/3 + 4. 2/3 of pi is about 1 less than pi. 4 is about 0.9 more than pi. Add those two numbers together and the sum is about 0.1 less than 2 pi. The sine of 2 pi = sin of zero = zero. The slope of the sine function at the origin is 1. But here it's slightly less. 2cosine of zero is 2. But it's slightly less or around 1.9 Graph a few points to feel absolutely confident you have the right solution.
Maximum slope for the sin(x) is 1, which is the maximum value of cosine(x) But with sine of 2x, the period is halved, and the frequency doubled. The sine function is like a coil bunched up, so the maximum slope is now twice as much or 2. Slightly less or more and the slope is about 1.9
