Unit 5: Analytical Applications of Differentiation

This unit uses derivatives to analyze the full shape of a function: where it rises or falls, where it bends, where extrema occur, and how to justify conclusions rigorously.

Critical points

A critical point of \(f\) occurs at \(x=c\) where:

\(f'(c) = 0\), or
\(f'(c)\) does not exist,

provided \(c\) is in the domain of \(f\).

Increasing and decreasing

\(f'(x) > 0\) on an interval implies \(f\) is increasing there.
\(f'(x) < 0\) on an interval implies \(f\) is decreasing there.

Sign charts are the cleanest way to justify interval behavior.

First Derivative Test

If \(f'\) changes:

positive to negative at \(c\): local maximum,
negative to positive at \(c\): local minimum,
no sign change: neither.

Concavity and second derivative

\(f''(x) > 0\) means \(f\) is concave up.
\(f''(x) < 0\) means \(f\) is concave down.

An inflection point is a point where concavity changes.

[Image Placeholder: graph showing local extrema and inflection points with sign charts]

Second Derivative Test

If \(f'(c)=0\) and:

\(f''(c)>0\), then \(f\) has a local minimum at \(c\),
\(f''(c)<0\), then \(f\) has a local maximum at \(c\),
\(f''(c)=0\), the test is inconclusive.

Absolute extrema on a closed interval

To find absolute max/min of \(f\) on \([a,b]\):

Find critical points inside \((a,b)\).
Evaluate \(f\) at each critical point.
Evaluate \(f(a)\) and \(f(b)\).
Compare all values.

Mean Value Theorem

If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there exists \(c \in (a,b)\) such that

\[f'(c) = \frac{f(b)-f(a)}{b-a}.\]

Rolle’s Theorem is the special case where \(f(a)=f(b)\).

Curve sketching framework

A solid derivative-based sketch includes:

intercepts,
asymptotes if relevant,
intervals increasing/decreasing,
local extrema,
intervals concave up/down,
inflection points,
end behavior.

Optimization

Standard process:

Identify the quantity to optimize.
Write it as a function of one variable.
Determine the feasible domain.
Differentiate and find critical points.
Test candidates and interpret.

L’Hopital’s Rule

If a limit produces \(0/0\) or \(\infty/\infty\) and the hypotheses are satisfied, then

\[\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\]

provided the new limit exists in a usable way.

Linearization and Newton’s method

Linearization:

\[L(x) = f(a)+f'(a)(x-a).\]

Newton’s method for approximating roots:

\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.\]

[Image Placeholder: Newton’s method tangent-line iteration toward a root]

Common mistakes

Calling every critical point an extremum.
Using the second derivative test when \(f'(c) \ne 0\).
Forgetting endpoints in absolute-extrema problems.
Claiming an inflection point from \(f''=0\) without checking concavity change.