Calculus I Exam 1 Review: Limits, Continuity, and Derivatives

Limit Basics

Definition and Interpretation of Limits

The concept of a limit is foundational in calculus, describing the behavior of a function as its input approaches a particular value.

• Limit Notation: $\lim_{x \to a} f(x) = L$ means that as $x$ approaches $a$, $f(x)$ approaches $L$.

• Key Points:

  • The limit describes what $f(x)$ approaches as $x$ gets close to $a$, not necessarily the value at $x = a$.

  • $\lim_{x \to a} f(x) = L$ does not guarantee that $f(a) = L$; it only describes the behavior near $a$.

• One-Sided Limits: Be familiar with left-hand ($\lim_{x \to a^-} f(x)$) and right-hand ($\lim_{x \to a^+} f(x)$) limits.

• Limit Existence: $\lim_{x \to a} f(x) = L$ only if both one-sided limits exist and are equal to $L$.

• Graphical Interpretation: Be able to use a graph to answer questions about limits.

Examples

• Example: For $f(x) = \frac{x^2 - 1}{x - 1}$, as $x \to 1$, $f(x)$ approaches $2$even though$f(1)$ is undefined.

Continuity

Definition and Properties

Continuity describes a function that does not have any abrupt jumps, holes, or breaks at a point or over an interval.

• Definition: $f(x)$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.

• One-Sided Continuity: There is one-sided continuity if the one-sided limit equals the function value.

• Interval Continuity: $f(x)$ is continuous on an interval if it is continuous at every point in the interval.

• Common Points of Discontinuity:

  • Division by zero

  • Even roots of negative numbers

  • Logarithms of non-positive numbers

  • Piecewise rule changes or undefined points

Examples

• Example: $f(x) = \frac{1}{x}$ is discontinuous at $x = 0$ due to division by zero.

Limit Evaluation Techniques

Algebraic Methods

Limits can often be evaluated using algebraic manipulation, especially when direct substitution leads to indeterminate forms.

• Factoring: If the limit is of the form $\frac{0}{0}$, try to factor numerator and denominator and simplify.

• Multiplying by Conjugate: For limits involving square roots, multiply by the conjugate to simplify.

• Handling Fractions: Know how to handle limits that involve complex fractions.

• One-Sided Limits: For forms like $\frac{c}{0}$, consider the direction of approach and sign.

• Notation: Be careful not to drop the limit operator until the operation is complete.

Examples

• Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$: Factor numerator to get $\frac{(x-2)(x+2)}{x-2}$, so the limit is $4$.

Limit Definition of the Derivative

Concept and Application

The derivative of a function at a point measures the instantaneous rate of change, defined as the limit of the difference quotient.

• Definition:

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

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

• Geometric Interpretation: The derivative gives the slope of the tangent line to $y = f(x)$ at $x = a$.

• Secant and Tangent Lines: The slope of the secant line approximates the average rate of change; the tangent line gives the instantaneous rate.

• Equation of Tangent Line: $y = f'(a)(x - a) + f(a)$

Examples

• Example: For $f(x) = x^2$, $f'(a) = \lim_{h \to 0} \frac{(a+h)^2 - a^2}{h} = 2a$.

Derivative Rules

Basic Derivative Formulas

Knowing the derivatives of basic functions and how to apply rules is essential for differentiation.

• Constant Rule: $\frac{d}{dx} C = 0$

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

• Exponential Rule: $\frac{d}{dx} e^x = e^x$

• Logarithmic Rule: $\frac{d}{dx} \ln x = \frac{1}{x}$

• Trigonometric Rules:

  • $\frac{d}{dx} \sin x = \cos x$

  • $\frac{d}{dx} \cos x = -\sin x$

  • $\frac{d}{dx} \tan x = \sec^2 x$

  • $\frac{d}{dx} \cot x = -\csc^2 x$

  • $\frac{d}{dx} \sec x = \sec x \tan x$

  • $\frac{d}{dx} \csc x = -\csc x \cot x$

• Inverse Trigonometric Rules:

  • $\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}$

  • $\frac{d}{dx} \arctan x = \frac{1}{1 + x^2}$

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

• Constant Multiple Rule: $\frac{d}{dx} [k f(x)] = k f'(x)$

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

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

Exponential and Logarithmic Rules

• Exponential Rule:

  • $\frac{d}{dx} a^x = a^x \ln a$

  • $\frac{d}{dx} e^{kx} = k e^{kx}$

• Logarithmic Rules:

  • $\ln(ab) = \ln a + \ln b$

  • $\ln \left( \frac{a}{b} \right) = \ln a - \ln b$

  • $\ln(a^r) = r \ln a$

Implicit Differentiation

Concept and Application

Implicit differentiation is used when a function is not given explicitly as $y = f(x)$, but rather in a form involving both $x$ and $y$.

• Solving for $\frac{dy}{dx}$: Differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$.

• Finding Slopes: $\frac{dy}{dx}$ gives the slope at all points on the curve; use it to find the slope at a specific point.

• Horizontal/Vertical Tangents: Set $\frac{dy}{dx} = 0$ for horizontal tangents, and solve for vertical tangents where the denominator is zero.

• Logarithmic Differentiation: Use logarithmic differentiation for functions of the form $f(x)^{g(x)}$.

• Know the log and exponential rules.

Example

• Example: For $x^2 + y^2 = 1$, differentiate both sides to get $2x + 2y \frac{dy}{dx} = 0$, so $\frac{dy}{dx} = -\frac{x}{y}$.

Derivative Table

Common Derivatives

The following table summarizes the derivatives of common functions and rules:

Function	Derivative
$C$	$0$
$x$	$1$
$x^n$	$n x^{n-1}$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$
$\arcsin x$	$\frac{1}{\sqrt{1 - x^2}}$
$\arctan x$	$\frac{1}{1 + x^2}$
$k f(x)$	$k f'(x)$
$f(x) + g(x)$	$f'(x) + g'(x)$
$f(x) g(x)$	$f'(x) g(x) + f(x) g'(x)$
$\frac{f(x)}{g(x)}$	$\frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}$

Exponential and Logarithmic Rules Table

Rule	Formula
Exponential Rule	$\frac{d}{dx} a^x = a^x \ln a$
Exponential Rule	$\frac{d}{dx} e^{kx} = k e^{kx}$
Log Rule	$\ln(ab) = \ln a + \ln b$
Log Rule	$\ln \left( \frac{a}{b} \right) = \ln a - \ln b$
Log Rule	$\ln(a^r) = r \ln a$

Additional info:

• Some notation and examples have been expanded for clarity and completeness.

• Tables have been reconstructed to summarize key derivative and logarithmic rules.