Mathematics

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. a. Part 1: (a) Find the value of

k

that makes

s

continuous at

x=0

. Show your reasoning. b. 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}]

. c. 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. d. 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.