Calculus I Syllabus and Core Concepts Study Guide

Calculus I: Course Overview and Core Topics

Course Description

This course covers foundational concepts in calculus, including limits, continuity, derivatives, and integrals. Students will learn techniques of differentiation and integration, and apply these concepts to solve problems involving maxima, minima, rates of change, and areas under curves. The course emphasizes both theoretical understanding and practical applications, with a focus on algebraic, trigonometric, and transcendental functions.

Core Calculus Topics

Limits and Continuity

Limits are fundamental to calculus, providing a way to describe the behavior of functions as inputs approach specific values. Continuity ensures that functions behave predictably without sudden jumps or breaks.

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

• Left and Right-Hand Limits: Limits as the input approaches from the left ($x \to a^-$) or right ($x \to a^+$).

• Infinite Limits: Describes behavior as the function grows without bound near a point.

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

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

• 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 $f(c) = N$.

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

Example: Find $\lim_{x \to 2} (x^2 - 4)/(x - 2)$. Factor numerator: $x^2 - 4 = (x - 2)(x + 2)$, so limit is $x + 2$ as $x \to 2$, giving $4$.

Derivatives and Differentiation Rules

The derivative measures the instantaneous rate of change of a function. Differentiation rules allow efficient computation of derivatives for various types of functions.

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

• Derivative Notations: $f'(x)$, $\frac{df}{dx}$, $\frac{d}{dx}f(x)$

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

• Product Rule: $\frac{d}{dx}[u \cdot v] = u'v + uv'$

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

• 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 $y$.

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

• Related Rates: Application of derivatives to problems where multiple variables change with respect to time.

Example: If $y = x^3$, then $y' = 3x^2$.

Applications of the Derivative

Derivatives are used to analyze the behavior of functions, including finding maxima and minima, understanding the shape of graphs, and solving optimization problems.

• Maxima and Minima: Points where a function reaches its highest or lowest value locally or globally.

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

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

• First Derivative Test: Used to determine local maxima and minima.

• Second Derivative Test: Used to determine concavity and inflection points.

• L'Hopital'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).

• Optimization: Using derivatives to find maximum or minimum values in applied problems.

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

Example: To maximize $f(x) = -x^2 + 4x$, set $f'(x) = -2x + 4 = 0 \Rightarrow x = 2$.

Integration and Applications

Integration is the process of finding the area under a curve or accumulating quantities. The Fundamental Theorem of Calculus connects differentiation and integration.

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

• Indefinite Integral: $\int f(x)\,dx$ gives the family of antiderivatives of $f(x)$.

• Riemann Sum: $\sum_{i=1}^n f(x_i^*) \Delta x$ approximates area under $f(x)$.

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

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

Example: $\int_0^2 x\,dx = \frac{1}{2}x^2 \Big|_0^2 = 2$

Course Structure and Evaluation

Grading System

The course grade is determined by homework, labs, module exams, and a final exam. The breakdown is as follows:

Type	Weight	Notes
Homework	10%	Completed on Pearson MyLab
Labs and CORE Assessments	10%	Completed on paper
Module Exams	60%	Completed on paper in class
Final Exam	20%	Comprehensive, completed on paper in class

Grade Scale:

Grade	Overall Average
A	89.5% to 100%
B	79.5% to 89.4%
C	69.5% to 79.4%
D	59.5% to 69.4%
F	< 59.5%

Course Calendar: Key Topics by Week

The course is organized into modules, each focusing on a major calculus topic:

• Module 1: Limits and Continuity

• Module 2: Derivatives and Differentiation Rules

• Module 3: Applications of Differentiation

• Module 4: Integrals

Each module includes homework, labs, and assessments to reinforce learning.

Student Learning Outcomes

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

• Draw graphs of algebraic and transcendental functions considering limits, continuity, and differentiability.

• Determine continuity and differentiability at a point using limits.

• Use differentiation rules to differentiate algebraic and transcendental functions.

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

• Evaluate definite integrals using the Fundamental Theorem of Calculus.

• Articulate the relationship between derivatives and integrals.

Additional Support and Resources

• Math Lab: Free tutoring available for calculus students.

• Graphing Calculator: TI-83 or TI-84 recommended.

• Textbook: Calculus Early Transcendentals by Briggs, Pearson, 3rd Edition.

Summary Table: Major Calculus Concepts

Concept	Definition	Key Formula
Limit	Value function approaches as input nears a point	$\lim_{x \to a} f(x)$
Derivative	Instantaneous rate of change	$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
Definite Integral	Area under curve between two points	$\int_a^b f(x)\,dx$
Fundamental Theorem of Calculus	Connects differentiation and integration	$\int_a^b f(x)\,dx = F(b) - F(a)$
Chain Rule	Derivative of composite functions	$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$
L'Hopital's Rule	Solves indeterminate forms	$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

Additional info: These notes are based on the syllabus and module outcomes for Calculus I at Collin College, Spring 2026. They provide a structured overview of the main topics, definitions, and formulas students will encounter in the course.