Calculus I: Limits, Continuity, and Derivatives – Study Guide

Limits and Continuity

Understanding Limits

Limits are fundamental to calculus, describing the behavior of a function as the input approaches a particular value. They are essential for defining derivatives and integrals.

• Limit Notation: $\lim_{x \to a} f(x)$ denotes the value that f(x) approaches as x approaches a.

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

• Existence of Limits: The limit exists at x = a if and only if both one-sided limits exist and are equal.

• Limits at Infinity: Describes the behavior of a function as x grows without bound.

Example: Given a graph of f(x), to find $\lim_{x \to 2} f(x)$, observe the y-value the function approaches as x approaches 2 from both sides.

Continuity

A function is continuous at a point if its limit at that point exists and equals the function's value there.

• Continuity Checklist at x = a:

  1. f(a) is defined.

  2. $\lim_{x \to a} f(x)$ exists.

  3. $\lim_{x \to a} f(x) = f(a)$.

• Types of Discontinuities: Removable, jump, and infinite (essential).

Example: For a piecewise function, check each condition at the point where the formula changes.

Evaluating Limits from Graphs and Functions

Graphical Approach

To evaluate limits from a graph:

• Trace the curve as x approaches the target value from both sides.

• If the left and right approaches yield the same y-value, the limit exists and equals that value.

• If the function jumps or diverges, the limit does not exist (DNE).

Example: If the graph of f(x) has a hole at x = 3 but the curve approaches the same y-value from both sides, the limit exists even if f(3) is undefined.

Algebraic Approach

• Direct substitution: Plug in the value if the function is defined and continuous there.

• Factor and cancel: For rational functions with indeterminate forms (like 0/0), factor numerator and denominator and simplify.

• Use conjugates: For limits involving square roots, multiply by the conjugate to simplify.

• Special limits: Recognize standard limits such as $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

Piecewise Functions and Continuity

Defining Continuity for Piecewise Functions

For a piecewise function, continuity at the point where the formula changes requires matching the left and right limits and the function value.

• Set $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$ and solve for any unknowns.

Example: For $f(x) = \begin{cases} kx+1, & x \leq 3 \\ 2x-k, & x > 3 \end{cases}$, set $\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x)$ to solve for k.

The Derivative: Definition and Interpretation

Definition of the Derivative

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

• Limit Definition:

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

  • Alternatively, $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

• Geometric Interpretation: Slope of the tangent line to the curve at x = a.

• Physical Interpretation: Instantaneous velocity if f(x) represents position.

Derivative from a Table or Data

• Average velocity: $\frac{\Delta s}{\Delta t}$ over an interval.

• Estimate instantaneous velocity by taking average velocities over smaller intervals around the point of interest.

Example: If $s(t)$ is position at time t, then $v(t) = s'(t)$ is velocity, and $a(t) = v'(t)$ is acceleration.

Basic Derivative Rules

Common Derivatives

• $\frac{d}{dx} x^n = n x^{n-1}$

• $\frac{d}{dx} e^x = e^x$

• $\frac{d}{dx} \ln x = \frac{1}{x}$

• $\frac{d}{dx} \sin x = \cos x$

• $\frac{d}{dx} \cos x = -\sin x$

• $\frac{d}{dx} a^x = a^x \ln a$

• $\frac{d}{dx} \log_a x = \frac{1}{x \ln a}$

Product, Quotient, and Chain Rules

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

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

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

Applications of Derivatives

Tangent Lines

The equation of the tangent line to y = f(x) at x = a is:

• $y = f(a) + f'(a)(x - a)$

Example: To find the tangent to a curve at a point, compute the derivative at that point and use the point-slope form.

Motion Along a Line

• Position function: $s(t)$

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = v'(t) = s''(t)$

• Object is at rest when $v(t) = 0$.

• Object changes direction when $v(t)$ changes sign.

Example: For $s(t) = 2t^3 - 21t^2 + 108t + 53$, find when $v(t) = 0$ to determine when the particle is at rest.

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to another, often using implicit differentiation.

• Differentiate both sides of an equation with respect to time t.

• Substitute known values to solve for the desired rate.

Example: If a ladder slides down a wall, relate the rates of change of the height and the distance from the wall using the Pythagorean theorem.

Logarithmic Differentiation

Logarithmic differentiation is useful for functions of the form $y = f(x)^{g(x)}$ or products/quotients of many functions.

• Take the natural logarithm of both sides: $\ln y = \ln(f(x)^{g(x)})$

• Differentiate both sides using the chain rule and properties of logarithms.

Table: Derivative and Limit Properties

Function	Derivative
$x^n$	$n x^{n-1}$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$a^x$	$a^x \ln a$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$

Intermediate Value Theorem (IVT)

The IVT 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 the existence of roots or values within an interval.

Example: If f(2) = -5 and f(5) = 5, then for any value between -5 and 5, there is some c in (2, 5) with f(c) = N.

Summary Table: Types of Discontinuities

Type	Description	Example
Removable	Hole in the graph; limit exists, but function value is missing or different	$f(x) = \frac{x^2-1}{x-1}$ at $x=1$
Jump	Left and right limits are different	Piecewise function with different values at a point
Infinite	Function approaches infinity	$f(x) = \frac{1}{x}$ at $x=0$

Additional info: Some explanations and examples have been expanded for clarity and completeness, as the original file contained only brief questions and answers.