Applications of differentiation

Question

f(x)= sin(x)+1 for the closed interval [0, pi/2]

find the value of x to at least 3 decimal places where the average rate of change of the function equals the instantaneous rate of change of the function on this interval.  

 

I'm confused on this problem, and its driving me crazy because I have found the average rate of change but have not been able to find the instantaneous rate of change!!

Answer 1

Emma,

 

You are so much closer to the answer than you think you are! 

 

I assume that you found the average rate of change by finding the slope over the given interval. That is (2 - 1)/(pi/2 - 0) = 2/pi

 

Now we use the "mean value theorem" which says that the average slope is equal to the slope at some specific x value. The key word here is "the slope of a specific value", which means we need to find the derivative of f(x) when the slope is 2/pi

 

Fist, we find the derivative of the given function.

 

f'(x) = cos(x) + 0

 

2/pi = cos(x)

x = cos^-1(2/pi)

x = 0.8806 radians

x = 50.5 degrees

 

I hope this helps!