Calculus I Study Guide: Limits, Derivatives, Tangent Lines, and Asymptotes

Limits and Average Rate of Change

Secant Lines and Average Rate of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input. In the context of motion, it represents the average speed over a time interval.

• Secant Line: A line passing through two points on a curve. Its slope gives the average rate of change between those points.

• Formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$ where $f(a)$ and $f(b)$ are function values at $a$ and $b$.

• Example: If a runner covers 1200 meters in 10 minutes and 2700 meters in 30 minutes, the average speed between 10 and 30 minutes is: $\frac{2700 - 1200}{30 - 10} = \frac{1500}{20} = 75$ meters/minute.

Instantaneous Rate of Change and Tangent Lines

The instantaneous rate of change at a point is the slope of the tangent line to the curve at that point. It is found using derivatives.

• Tangent Line: A line that touches the curve at one point and has the same slope as the curve at that point.

• Derivative: The limit of the average rate of change as the interval shrinks to zero.

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

Limits and Continuity

Definition of a Limit

The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ gets closer to as $x$ gets closer to $a$.

• Notation: $\lim_{x \to a} f(x)$

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

• Existence: The limit exists if both one-sided limits are equal.

• Example: For $f(x) = x-2$ for $x \leq 3$, $f(x) = 2x+7$ for $x > 3$, $\lim_{x \to 3} f(x) = 1$ if both sides approach 1.

Evaluating Limits

• Direct Substitution: Plug in the value of $x$ if the function is continuous at that point.

• Factoring: Factor numerator and denominator to cancel common terms.

• Rationalization: Multiply by conjugate to simplify expressions with roots.

• Special Limits: Use known limits such as $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

• Example: $\lim_{x \to 0} \sqrt{x^3 + x^2} \sin \frac{1}{x}$

Continuity

A function is continuous at $x=a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

Asymptotes and Holes

Vertical and Horizontal Asymptotes

An asymptote is a line that a graph approaches but never touches.

• Vertical Asymptote: Occurs when the denominator of a rational function is zero and the numerator is not zero at that point. Example: For $f(x) = \frac{3x+1}{x^2-4}$, vertical asymptotes at $x=2$ and $x=-2$.

• Horizontal Asymptote: Determined by the degrees of numerator and denominator. Example: If degrees are equal, horizontal asymptote at $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.

Holes in Graphs

A hole occurs when both numerator and denominator are zero at the same $x$-value, and the factor cancels.

• Example: $f(x) = \frac{x^2-4}{x-2}$ has a hole at $x=2$.

Derivatives and Tangent Lines

Definition of Derivative

The derivative of a function at a point measures the instantaneous rate of change, or the slope of the tangent line.

• Limit Definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Sum Rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$

• Example: $f(x) = x^2 + 3x$, $f'(x) = 2x + 3$

Finding the Equation of a Tangent Line

• Find the derivative $f'(x)$

• Evaluate $f'(a)$ at the point $x=a$

• Use point-slope form: $y - f(a) = f'(a)(x - a)$

• Example: For $y = x^2 + 3x$ at $(-1, -2)$, slope is $1$, equation is$y + 2 = 1(x + 1)$

Intermediate Value Theorem

Statement and Application

The Intermediate Value Theorem states that if $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c$ in $(a, b)$ such that $f(c) = N$.

• Used to show existence of roots in an interval.

• Example: If $f(-5) < 0$ and $f(0) > 0$, then $f(x)$ has a root in $(-5, 0)$.

Domain of Functions

Finding the Domain

The domain of a function is the set of all input values for which the function is defined.

• Exclude values that make the denominator zero or result in negative values under even roots.

• Example: For $g(x) = \sqrt{x-3}$, domain is $x \geq 3$.

Summary Table: Types of Asymptotes

Type	How to Find	Example
Vertical Asymptote	Set denominator = 0, solve for x	$x^2-4=0 \Rightarrow x=2, -2$
Horizontal Asymptote	Compare degrees of numerator and denominator	$y=0$ if degree numerator < denominator
Hole	Common factor cancels in numerator and denominator	$x^2-4$ and $x-2$ at $x=2$

Evaluating Limits Numerically

Numerical Substitution

• Plug in values close to the point of interest to estimate the limit.

• Example: $\lim_{x \to 5} \frac{x+2}{x-6}$, try $x=5.001$

Key Formulas and Rules

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Product Rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)$

Practice Problems and Applications

• Find the slope of the tangent line to $y = x^2 + 3x$ at $(-1, -2)$

• Find all asymptotes and holes for $f(x) = \frac{x^2-3x-4}{x^2-x}$

• Use the Intermediate Value Theorem to show a root exists for $f(x) = x^2 + 2x - 8$ in $[-5, 0]$

Additional info: Some explanations and examples have been expanded for clarity and completeness.