Mathematics

Problem

Define the function

h(x)=\begin{cases} \sin\!\left(\dfrac{\pi x}{4}\right), & x<2,\\ a, & 2\le x<2.5,\\ \ln(x+1), & x\ge 2.5, \end{cases}

where

a

is a constant. a. Part 1: (a) Evaluate

\lim_{x\to 2.5^-}h(x)

and

\lim_{x\to 2.5^+}h(x)

. Classify the discontinuity, if any, at

x=2.5

and compute its jump size. b. Part 2: (b) Let

J(a)

denote the jump size at

x=2

. Express

J(a)

in terms of

a

, and determine the value of

a

that minimizes

J(a)

. Show all reasoning. c. Part 3: (c) Using the value of

a

found in part (b), state the total sum of the jump sizes at

x=2

and

x=2.5

combined. Justify your calculation. d. Part 4: (d) Provide a qualitative sketch of

y=h(x)

for

0\le x\le 4

, clearly indicating the discontinuities and the behavior of the function on each interval.