Calculus Study Guide: Average Rate of Change, Limits, and Tangents

Average Rate of Change

Definition and Calculation

The average rate of change of a function over an interval [a, b] measures how much the function's output changes per unit change in input. It is calculated as:

• Formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

• Interpretation: Represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)) on the graph.

Example: For $h(t) = \cot t$ over $\left[\frac{3\pi}{4}, \pi\right]$:

• $\frac{\cot(\pi) - \cot\left(\frac{3\pi}{4}\right)}{\pi - \frac{3\pi}{4}} = \frac{-1 - 0}{\frac{\pi}{4}} = -\frac{4}{\pi}$

Application: Used to estimate rates such as velocity, growth, or change in economics and sciences.

Limits and Their Properties

Definition of a Limit

A limit describes the value that a function approaches as the input approaches a certain point. Limits are foundational in calculus for defining continuity, derivatives, and integrals.

• Notation: $\lim_{x \to a} f(x)$

• Existence: The limit exists if the left-hand and right-hand limits are equal.

• One-Sided Limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right)

Example: $\lim_{x \to 0} \sec x = 1$

Evaluating Limits

• Direct Substitution: If $f(x)$ is continuous at $x=a$, then $\lim_{x \to a} f(x) = f(a)$.

• Factoring and Simplifying: Used when direct substitution yields an indeterminate form (e.g., $0/0$).

• Special Limits: Limits involving trigonometric, radical, or piecewise functions may require algebraic manipulation or known limit properties.

Examples:

• $\lim_{x \to 4} \frac{x^2 + x - 12}{x - 4} = -7$

• $\lim_{x \to 169} \frac{\sqrt{x} - 13}{x - 169} = \frac{1}{26}$

• $\lim_{x \to 5} \frac{\sqrt{x^2 + 11} - 6}{x - 5} = \frac{5}{6}$

• $\lim_{x \to 0} \frac{5 + 2x + \sin x}{6 \cos x} = \frac{5}{6}$

Limits Involving Trigonometric Functions

• Key Limit: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

• Example: $\lim_{x \to 0} \frac{x \cos 5x}{6 \cos 18x} = \frac{1}{5}$

• Example: $\lim_{\theta \to 0^+} \frac{\tan \theta}{\cot 6\theta} = 6$

Limits Involving Piecewise and Greatest Integer Functions

• Piecewise Functions: Limits may not exist at points where the function definition changes.

• Greatest Integer Function: $y = \lfloor x \rfloor$; limits from the left and right may differ at integer values.

• Example: $\lim_{x \to 2^-} \lfloor x \rfloor = 1$, $\lim_{x \to 2^+} \lfloor x \rfloor = 2$

Instantaneous Rate of Change and Tangents

Definition and Connection to Derivatives

The instantaneous rate of change at a point is the slope of the tangent line to the curve at that point. It is defined as:

• Formula: $\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Interpretation: This is the definition of the derivative $f'(x)$.

Example: For $f(x) = x^2$ at $x = 7$:

• $\lim_{h \to 0} \frac{(7 + h)^2 - 49}{h} = 14$

Equation of the Tangent Line

• General Form: $y = f'(a)(x - a) + f(a)$

• Example: For $y = 5 - 9x^2$ at $P(-4, -139)$, slope $= 72$, tangent line: $y = 72x + 149$

Continuity and Existence of Limits

Definition of Continuity

A function $f(x)$ is continuous at $x=a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

Discontinuities: Occur when any of the above conditions fail. Types include jump, infinite, and removable discontinuities.

Limits and Function Values

• Existence of $\lim_{x \to a} f(x)$ does not guarantee $f(a)$ is defined, and vice versa.

• If $f(a)$ is defined, $\lim_{x \to a} f(x)$ may still not exist (e.g., jump discontinuity).

Example: If $f(x)$ is piecewise, $\lim_{x \to 1} f(x)$ may not equal $f(1)$.

Graphical Interpretation of Limits and Continuity

Domain and Range

• Domain: The set of all input values for which the function is defined.

• Range: The set of all possible output values.

Example: For a given graph, domain $[0,6]$, range $(0,3) \cup \{6\}$.

Left-Hand and Right-Hand Limits

• Left-Hand Limit: $\lim_{x \to a^-} f(x)$

• Right-Hand Limit: $\lim_{x \to a^+} f(x)$

• Limits may exist from one side but not the other, especially at endpoints or discontinuities.

Special Limit Forms and Techniques

Indeterminate Forms

• Common indeterminate forms: $0/0$, $\infty/\infty$

• Techniques: Factoring, rationalizing, L'Hospital's Rule (in advanced calculus)

Limits Involving Radicals and Trigonometric Functions

• Example: $\lim_{x \to 19} \frac{\sqrt{x} - f(19)}{x - 19} = \frac{1}{2\sqrt{19}}$ for $f(x) = \sqrt{x}$

• Example: $\lim_{x \to 0} \frac{8 \sin \sqrt{9\theta}}{\sqrt{9\theta}} = 8$

Summary Table: Common Limit Results

Limit Expression	Result	Notes
$\lim_{x \to 0} \frac{\sin x}{x}$	1	Fundamental trigonometric limit
$\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$	$f'(a)$	Definition of derivative
$\lim_{x \to a} \frac{\sqrt{x} - \sqrt{a}}{x - a}$	$\frac{1}{2\sqrt{a}}$	Radical limit
$\lim_{x \to 0} \frac{1 - \cos \theta}{\sin 15\theta}$	0	Trigonometric limit

Additional info:

• Some questions involve graphical analysis, interval notation, and piecewise functions, which are essential for understanding limits and continuity in calculus.

• Questions cover both computational and conceptual aspects, including the existence of limits, left/right-hand limits, and the relationship between limits and function values.