Sec 6.5: Average Value of a Function

1. Find the average value of the function on the given interval.

(a) (i) \(f(x)=2x^2-3, \quad [1,3] \)   (ii) \(f(x)=x^2+2x-3, \quad [-1,2] \)   solution

(b) \( f(x)=1+3x^2, \quad [-2, 1] \)   solution

(c) \(f(x)=\cos (2x), \quad [-\pi, \pi] \)   solution

(d) \(f(x)=6x(1+x^2)^2, \quad [1,2] \)  solution

(e) \( \displaystyle{g(t)=\frac{t}{\sqrt{5+t^{2}}}, \quad[2,5] } \)   solution

(f) \(\displaystyle{f(x)=x\sqrt{25-x^2}, \quad [3, 5]} \)   solution

(g) \( \displaystyle{h(u)=\frac{\ln u}{u}, \quad[1, 7] } \)  solution


2.The temperature in \( { }^{\circ} \mathrm{F}\) in a city \(t\) hours after 9:00 AM is modelled by the function \( \displaystyle{T(t)= 50+8 14 \sin \left(\frac{\pi t}{12} \right) }\). Find the average temperature in that city in between 9:00 AM and 9:00 PM.  solution