Calculus I (MATH-2413) Syllabus and Core Concepts Study Guide

Course Overview

This study guide summarizes the key topics, learning outcomes, and structure of Calculus I (MATH-2413), as outlined in the course syllabus. The course covers foundational concepts in calculus, including limits, derivatives, applications of derivatives, and integration. The guide is organized by major topics and subtopics, providing definitions, examples, and essential formulas to support exam preparation and conceptual understanding.

Limits and Continuity

Introduction to Limits

Limits are fundamental to calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for defining derivatives and integrals.

• Definition of a Limit: The value that a function approaches as the input approaches a certain point.

• One-Sided Limits: Limits approached from the left ($x \to a^-$) or right ($x \to a^+$).

• Infinite Limits: When a function increases or decreases without bound as it approaches a point.

• Limits at Infinity: Describes the behavior of a function as $x \to \infty$ or $x \to -\infty$.

• Techniques for Computing Limits: Direct substitution, factoring, rationalization, and using limit laws.

• Squeeze Theorem: If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

• Precise (ε-δ) Definition of Limit: For every $\varepsilon > 0$, there exists $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \varepsilon$.

Example: $\lim_{x \to 2} (3x + 1) = 7$

Continuity

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

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

• Types of Discontinuities: Removable, jump, and infinite discontinuities.

• Intermediate Value Theorem: 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 $f(c) = N$.

Example: The function $f(x) = x^2$ is continuous everywhere.

Derivatives

Definition and Interpretation

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

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

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

• Differentiability: A function is differentiable at $x = a$ if the derivative exists at that point.

Example: For $f(x) = x^2$, $f'(x) = 2x$

Rules of Differentiation

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

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

• Derivatives of Trigonometric Functions: $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = -\sin x$

• Implicit Differentiation: Used when $y$ is defined implicitly by an equation involving $x$ and $y$.

• Logarithmic and Exponential Derivatives: $\frac{d}{dx} e^x = e^x$, $\frac{d}{dx} \ln x = \frac{1}{x}$

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

Example: $\frac{d}{dx} (x^3 \sin x) = 3x^2 \sin x + x^3 \cos x$

Applications of Derivatives

• Rates of Change: Derivatives represent velocity, acceleration, and other rates in applied problems.

• Related Rates: Problems involving two or more related quantities changing over time.

• Higher Order Derivatives: The second derivative $f''(x)$ gives information about concavity and acceleration.

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

Applications of the Derivative

Extrema and Optimization

Derivatives are used to find local and absolute maxima and minima of functions, which are critical in optimization problems.

• Critical Points: Points where $f'(x) = 0$ or $f'(x)$ does not exist.

• First Derivative Test: Determines if a critical point is a local maximum or minimum.

• Second Derivative Test: If $f''(c) > 0$, $f$ has a local minimum at $c$; if $f''(c) < 0$, a local maximum.

• Optimization: Using derivatives to solve real-world problems involving maximum or minimum values.

Example: Maximizing the area of a rectangle with a fixed perimeter.

Mean Value Theorem and Rolle's Theorem

• Mean Value Theorem: If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

• Rolle's Theorem: If $f(a) = f(b)$, then there exists $c \in (a, b)$ such that $f'(c) = 0$.

Curve Sketching and Concavity

• Increasing/Decreasing Intervals: Determined by the sign of $f'(x)$.

• Concavity and Inflection Points: $f''(x) > 0$ indicates concave up; $f''(x) < 0$ concave down. Inflection points occur where concavity changes.

• Graphing Functions: Use critical points, inflection points, and asymptotes to sketch graphs.

Other Applications

• Linear Approximations: $L(x) = f(a) + f'(a)(x - a)$ approximates $f(x)$ near $x = a$.

• Differentials: $dy = f'(x) dx$ estimates change in $y$ for small $dx$.

• L'Hospital's Rule: For indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$, $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the limit exists).

• Newton's Method: Iterative method for approximating roots: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Integration

Definite and Indefinite Integrals

Integration is the inverse process of differentiation and is used to find areas, accumulated quantities, and solve differential equations.

• Indefinite Integral: $\int f(x) dx = F(x) + C$, where $F'(x) = f(x)$

• Definite Integral: $\int_a^b f(x) dx$ gives the net area under $f(x)$ from $x = a$ to $x = b$

• Fundamental Theorem of Calculus: If $F$ is an antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$

• Properties of Integrals: Linearity, additivity over intervals, and comparison properties.

Example: $\int_0^2 x dx = 2$

Riemann Sums and Area

• Riemann Sums: Approximating area under a curve by summing areas of rectangles.

• Left, Right, and Midpoint Sums: Different methods for choosing sample points in subintervals.

• Area as a Limit: $\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$

Integration Techniques

• Substitution Rule: $\int f(g(x))g'(x) dx = \int f(u) du$ where $u = g(x)$

• Applications: Calculating velocity, net change, and area between curves.

Applications of Integration

• Velocity and Net Change: $\int_a^b v(t) dt$ gives the net change in position from $t = a$ to $t = b$

• Area Between Curves: $\int_a^b [f(x) - g(x)] dx$ where $f(x) \geq g(x)$ on $[a, b]$

Summary Table: Major Calculus I Concepts

Topic	Key Concepts	Representative Formula
Limits	One-sided, infinite, at infinity, Squeeze Theorem, ε-δ definition	$\lim_{x \to a} f(x)$
Continuity	Continuous at a point, types of discontinuities, Intermediate Value Theorem	$\lim_{x \to a} f(x) = f(a)$
Derivatives	Limit definition, rules, implicit, higher order, applications	$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
Applications of Derivatives	Extrema, optimization, Mean Value Theorem, curve sketching, L'Hospital's Rule	$f'(c) = \frac{f(b) - f(a)}{b - a}$
Integration	Definite/indefinite integrals, Riemann sums, substitution, area, net change	$\int_a^b f(x) dx = F(b) - F(a)$

Student Learning Outcomes (Summary)

• Develop solutions for tangent and area problems using limits, derivatives, and integrals.

• Draw and analyze graphs of algebraic and transcendental functions.

• Determine continuity and differentiability using limits.

• Apply differentiation rules to various functions.

• Model and solve real-world problems using calculus concepts.

• Evaluate definite integrals and articulate the relationship between derivatives and integrals.

Additional info: This guide is based on the course syllabus and learning outcomes for Calculus I (MATH-2413), and is intended to provide a structured overview of the main topics and skills required for success in the course.