Unizor - Derivatives - Exercises 2

Notes to a video lecture on http://www.unizor.com

Derivatives - Exercises 2

Exercise 2.1
Given a function f(x)=x·e ^-x.
Find an equation of a tangential line that touches this function at pointx=2
Answer
y = −e⁻²·x + 4·e⁻²


Exercise 2.2
Given a function f(x)=x·e^-x.
Find all its maximum, minimum and inflection points.
Answer
f ^I(x) = e^−x·(1−x)
f ^II(x) = e^−x·(x−2)
Point x=1 is a point of local maximum.
Point x=2 is a point of inflection.
There is no point of local minimum.

Exercise 2.3
Given a functionf(x)=sin(x)+cos(x).
Find all intervals where it's monotonically increasing and intervals where it's monotonically decreasing.
Answer
Intervals of monotonic increasing:
[−3π/4+2πn, π/4+2πn]
Intervals of monotonic decreasing:
[π/4+2πn, 5π/4+2πn]

Exercise 2.4
Given a function f(x)=sin(x).
Find all inflection points of this function.
What are the first derivatives at these points?
Answer
x = π·n
First derivatives equal to 1 or −1

Exercise 2.5
Given a function f(x)=x·e^-x.
Find an equation of a normal to its graph at point x=2
Answer
y = e²·x + 2·(e⁻²−e²)


Exercise 2.6
Given a sufficiently smooth function f(x).
Find equations of a tangential line and a normal to its graph at pointx=x₀
Answer
Tangential line:
y = f ^I(x₀)·(x−x₀) + f(x₀)
Normal:
y = −[f ^I(x₀)]⁻¹·(x−x₀) + f(x₀)