Derivatives and Critical Points Analysis

Derivatives

• First Derivative $f'(x)$: Reflects the slope or rate of change of $f(x)$ at any point $x$.
• Second Derivative $f''(x)$: Reflects the curvature or concavity of $f(x)$ at any point $x$.
• Higher Order Derivatives: The $n$-th derivative $f^{(n)}(x)$ represents the rate of change of the $(n-1)$-th derivative.

Critical Points

• A point $x = c$ is a critical point if $f'(c) = 0$ or if $f'(c)$ does not exist.
• Local Minimum: $f'(x) = 0$ and $f''(x) > 0$.
• Local Maximum: $f'(x) = 0$ and $f''(x) < 0$.
• Inflection Point: A point where $f''(x) = 0$ indicating a potential change in concavity.

Newton's Method

• An iterative algorithm to find successively better approximations to the roots of a real-valued function.
• Given by the formula: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

Implicit Differentiation

• Used for functions $y = g(x)$ that are not explicitly solved for $y$. The derivative $dy/dx$ is found using the chain rule and other differentiation rules.

Increasing and Decreasing Intervals

• Increasing: $f'(x) > 0$ implies that $f(x)$ is increasing on that interval.
• Decreasing: $f'(x) < 0$ implies that $f(x)$ is decreasing on that interval.

Concavity and Points of Inflection

• Concave Up: If $f''(x) > 0$ for all $x$ in an interval, $f(x)$ is concave up on that interval.
• Concave Down: If $f''(x) < 0$ for all $x$ in an interval, $f(x)$ is concave down on that interval.
• Inflection Point: If the sign of $f''(x)$ changes at $x = c$, then $c$ is an inflection point.

References

• tutorial.math.lamar.edu/pdf/calculus_cheat_sheet_derivatives.pdf
• personal.math.ubc.ca/~CLP/CLP1/clp_1_dc_text.pdf
• mitres_18_001_f17_full_book.pdf