Facts about derivatives

Meaning of Derivative

• Intuitively: instantaneous rate of change
• Geometrically: slope of tangent line to the graph

Calculation

• Simple functions: formulas
• Differentiation rules: sum, product, quotient, chain rule, etc.

Extrema

Definition

Let $f: I \to \mathbb{R}$ be a function defined on interval I.

• Absolute Maximum: If $f(x) \leq f(x_0)$ for all $x \in I$, then $f(x_0)$ is the absolute maximum
• Absolute Minimum: If $f(x) \geq f(x_0)$ for all $x \in I$, then $f(x_0)$ is the absolute minimum
• Relative Maximum: If there is an open interval $I' \subset I$ containing $x_0$ such that $f(x_0) \geq f(x)$ for all $x \in I'$, then $x_0$ is called a relative maximum
• Relative Minimum: If there is an open interval $I' \subset I$ containing $x_0$ such that $f(x_0) \leq f(x)$ for all $x \in I'$, then $x_0$ is called a relative minimum

Theorems

Weierstrass Theorem

A continuous function on a finite closed interval always has an absolute maximum and absolute minimum

Fermat’s Theorem

Let $x_0$ be a relative extremum of $f$, if $f'(x_0)$ exists then $f'(x_0) = 0$.

i.e.,

• A critical point is a point where $f'(x) = 0$ or undefined.
• Relative extrema can only occur at a critical point.
• But a critical point is NOT always a relative extremum.

Rolle’s Theorem

Conditions:

• $f$ is continuous on the closed interval $[a, b]$
• $f$ is differentiable on the open interval $(a, b)$
• $f(a) = f(b)$

Conclusion:

• There exists $c \in (a, b)$ such that $f'(c) = 0$

Geometric Interpretation:

• When endpoints are at equal heights, there must be a point where the tangent is horizontal.

Mean Value Theorem

Conditions:

• $f$ is continuous on the closed interval $[a, b]$
• $f$ is differentiable on the open interval $(a, b)$

Conclusion:

• There exists $c \in (a, b)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$

Geometric Interpretation:

• There exists a point where the slope of the tangent equals the slope of the secant line.

Determining

Relative Extrema: First Derivative Test

For a critical point $x$:

Absolute Extrema: Compare

• Find all critical points of $f$ in $(a, b)$
• Calculate the function values at critical points and endpoints
• Compare all function values
  • The largest value is the absolute maximum
  • The smallest value is the absolute minimum