A step function is defined by s(t)= cases 2, & 0 t<1

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

A step function

s

is defined by

s(t)=\begin{cases}2, & 0\le t<1,\\ t, & 1\le t<3,\\ 5, & t\ge 3.\end{cases}

An accumulation function

F

is given by

F(x)=\int_{0}^{x} s(t)\,dt,

for

x\ge 0

. (a) Compute

F(3)

. (b) Determine

F'(x)

for all

x>0

, expressing your answer as a piecewise-defined function. (c) Identify every value of

x

where

F''(x)

exists, and give its value when it exists. (d) Discuss the differentiability of

F

x=1

and

x=3

. Support your answer with appropriate derivative information.