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Exercise 2
Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007-2020)
Chapter 1: Differentiation; Section 3: Derivatives; page 2
Properties, Exercise 2
Question
Study and do approximate graphic presentation of the following function. The study should contain:
- Roots and infinities
- Increasing and decreasing intervals
- Stationary points and their characteristics
- Points of infinite derivative and their characteristics
- Inflection points
Parts
- There are two roots: x=0 and x=1 .
- The derivative is
- There is one stationary point
- There is one point with infinite derivative
which is also a root.
- At that point
The function is increasing at the left side
and decreasing at the right,
therefore the function has a maximum there.
- The second derivative is
- From this result it follows that at the stationary point (part 3) there is a minimum
- The function is
increasing for x<0 ,
decreasing for 0<x<0.4 and
increasing for x>0.4 .
Score
Parts 2 and 6 are worth two points each.
The rest (parts 1,3,4,5,7,8) are worth 1 point each.