Curve Sketching Examples

Tags: #◇Computer Science #◇CSMATH1 #◇Problem Solving

---

◇Overview

Curve sketching using derivatives allows us to determine the shape, direction, and critical features of a function's graph. By analyzing the first and second derivatives, we can find intervals of increase and decrease, relative extrema, concavity, and points of inflection.

◇Important Formulas & Concepts

Increasing and Decreasing Functions

Increasing Function: A function $y = f(x)$ is increasing on an interval $I$ if for any two points $x_1$ and $x_2$ in $I$, $x_1 < x_2 \implies f(x_1) \le f(x_2)$.
Decreasing Function: A function $y = f(x)$ is decreasing on an interval $I$ if for any two points $x_1$ and $x_2$ in $I$, $x_1 < x_2 \implies f(x_1) \ge f(x_2)$.
Monotonic Function: A function that is exclusively increasing or exclusively decreasing on an interval $I$.

The First Derivative Test for Monotonicity

1. If $f'(x) > 0$ for all $x$ in an interval $I$, then $f$ is increasing on $I$.
2. If $f'(x) < 0$ for all $x$ in an interval $I$, then $f$ is decreasing on $I$.

Critical Numbers and Points

Critical Number: A number $c$ in the domain of $f$ such that $f'(c) = 0$ or $f'(c)$ is undefined (d.n.e).
Critical Point: The point $(c, f(c))$ on the graph of $f$. At a critical point, the tangent line is either horizontal ($f'(c) = 0$) or there might be a sharp turn/cusp ($f'(c)$ is undefined).

First Derivative Test for Relative Extrema

Let $c$ be a critical number for $f$.

1. If $f'(x) > 0$ for $x < c$ and $f'(x) < 0$ for $x > c$, then $f$ has a relative maximum at $(c, f(c))$.
2. If $f'(x) < 0$ for $x < c$ and $f'(x) > 0$ for $x > c$, then $f$ has a relative minimum at $(c, f(c))$.
3. If $f'(x)$ does not change sign at $c$, then $f$ has no relative extremum at $c$.

(Note: A relative extremum point is necessarily a critical point, but a critical point may or may not be a relative extremum point. You must test the critical points.)

◇Worked Examples

Example: Find critical numbers and test for relative extrema.
Let $f(x)$ be a differentiable function.

1. Compute the first derivative $f'(x)$.
2. Set $f'(x) = 0$ and solve for $x$ to find critical numbers. Find any $x$ where $f'(x)$ is undefined.
3. Set up intervals based on the critical numbers to test the sign of $f'(x)$.
4. Apply the First Derivative Test to conclude whether each critical number corresponds to a relative maximum, relative minimum, or neither.

◇References & Navigation

• Previous Topic: ◇Absolute Extremum
• Next Topic: ◇Additional Slides
• Related Notes: ◇Functions and Their Graphs, ◇Absolute Extremum