Consider the piecewise function f(x)= cases x, & 0 x<1, 3, & x=1, 2-x, & 1< x 2

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Consider the piecewise function

f(x)=\begin{cases} x, & 0\le x<1,\\ 3, & x=1,\\ 2-x, & 1< x\le 2. \end{cases}

The interval of interest is

[0,2]

. (1) Part 1: (a) Investigate the continuity of

f

x=1

and classify any discontinuity present. (2) Part 2: (b) Explain whether the value of

f

at the single point

x=1

affects the existence or numerical value of the average value of

f

[0,2]

. Provide reasoning based on properties of definite integrals. (3) Part 3: (c) Compute the average value of

f

[0,2]

. Show all work.