*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 point

*x=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

**is a point of**

*x=1**local maximum*.

Point

**is a point of**

*x=2**inflection*.

There is no point of local

*minimum*.

*Exercise 2.3*

Given a function

**.**

*f(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

**or**

*1*

*−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 = e*^{2}·x + 2·(e^{−2}−e^{2})*Exercise 2.6*

Given a sufficiently smooth function

**.**

*f(x)*Find equations of a tangential line and a normal to its graph at point

*x=x*_{0}*Answer*

Tangential line:

*y = f*^{ I}(x_{0})·(x−x_{0}) + f(x_{0})Normal:

*y = −*[*f*]^{ I}(x_{0})^{−1}·(x−x_{0}) + f(x_{0})
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