Unizor - Derivatives - Easy Problems

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

Derivatives - Easy Problems

We recommend to go through these easy problems before watching the lecture with their solutions.
We offer a simple proof of the first theorem as a sample.

Theorem 1

Assume, function f(x) is differentiable (that is, has a derivative) at some point x0.
Symbolically, the following limit exists and equals to some constant K:
limx→x0[f(x)−f(x0)]/(x−x0) = K
or, in a short form, setting
f(x)−f(x0) = Δf(x) and
x−x0 = Δx,
this looks as follows
limΔx→0[Δf(x)/Δx] = K
Prove that it is continuous at this point.

Proof

Given:
Δf(x)/Δx → K as Δx → 0.
Therefore,
β(x) = Δf(x)/Δx − K
is infinitesimal variable
as Δx → 0.
From this we derive that
Δf(x) = [K+β(x)]·Δx
is also an infinitesimal if Δx → 0.
This is a definition of continuity of function f(x) at point x0.

Theorem 2

Prove that monotonically increasing in some interval differentiable function has non-negative derivative in this interval.

Theorem 3

Prove that monotonically decreasing in some interval differentiable function has non-positive derivative in this interval.

Theorem 4

Prove that, if a derivative of a differentiable function is positive in some interval, the function is monotonically increasing in this interval.

Theorem 5

Prove that, if a derivative of a differentiable function is negative in some interval, the function is monotonically decreasing in this interval.