Define the function on by s(x)= cases e^ x , & -2 x<0

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Define the function

s

[-2,\tfrac{\pi}{2}]

s(x)=\begin{cases} e^{x}, & -2\le x<0,\\ k, & x=0,\\ \cos x, & 0< x\le \tfrac{\pi}{2}, \end{cases}

where

k

is a real constant. Answer the following questions. (1) Part 1: (a) Find the value of

k

that makes

s

continuous at

x=0

. Show your reasoning. (2) Part 2: (b) For the value of

k

you found in part (a), calculate the average value of

s

on the interval

[-2,\tfrac{\pi}{2}]

. (3) Part 3: (c) Determine all numbers

c

[-2,\tfrac{\pi}{2}]

such that

s(c)

equals the average value computed in part (b). Justify your answer algebraically. (4) Part 4: (d) Discuss whether the average value of

s

[-2,\tfrac{\pi}{2}]

would exist if

k

were not chosen as in part (a). Support your conclusion.