MAT 112 Calculus I: Course Outline and Key Topics

Course Structure and Key Topics in Calculus I

Overview

This syllabus outlines the progression of topics in a college-level Calculus I course, including foundational concepts, differentiation, applications, and introductory integration. The schedule provides a logical sequence for learning and reviewing major calculus concepts.

1. Functions and Pre-Calculus Review

• Definition of a Function: A function is a relation that assigns each input exactly one output.

• Types of Functions: Linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions.

• Domain and Range: The set of possible inputs (domain) and outputs (range) for a function.

• Example: $f(x) = x^2$ is a quadratic function with domain $(- , )$ and range $[0, )$.

2. Limits and Continuity

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

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

• Key Formula: $\lim_{x \to a} f(x) = L$

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

3. The Derivative

• Definition: The derivative measures the rate at which a function changes as its input changes.

• Notation: $f'(x)$ or $\frac{df}{dx}$

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

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

4. Calculating Derivatives

• Derivative Rules: Power rule, product rule, quotient rule, and chain rule.

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

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

5. Applications of the Derivative

• Increasing/Decreasing Functions: Use the sign of $f'(x)$ to determine intervals of increase or decrease.

• Relative Extrema: Points where $f'(x) = 0$ and the function changes direction.

• Optimization: Finding maximum or minimum values of functions in applied contexts.

• Related Rates: Problems involving rates at which related quantities change.

• Example: Maximizing area given a fixed perimeter.

6. Curve Sketching and Higher Derivatives

• Curve Sketching: Use first and second derivatives to analyze and sketch graphs of functions.

• Concavity and Inflection Points: Determined by the sign of $f''(x)$.

• Higher Derivatives: Successive derivatives, such as $f''(x)$, $f'''(x)$, etc.

• Example: $f(x) = x^4$, $f'(x) = 4x^3$, $f''(x) = 12x^2$

7. Integration

• Antiderivative: A function whose derivative is the given function.

• Definite and Indefinite Integrals: Indefinite integrals represent families of functions; definite integrals compute area under curves.

• Key Formula: $\int f(x) dx$

• Fundamental Theorem of Calculus: Relates differentiation and integration.

• Example: $\int x^2 dx = \frac{1}{3}x^3 + C$

8. Techniques of Integration

• U-Substitution: Used to simplify integrals by substituting variables.

• Integration by Parts: Based on the product rule for derivatives.

• Area Between Curves: Calculated using definite integrals.

• Example: $\int 2x \cos(x^2) dx$ (use $u = x^2$)

9. Review and Exam Preparation

• Regular Review: Scheduled review sessions for chapters and special topics.

• Midterm and Final Exam: Comprehensive assessments covering all course material.

Course Schedule Table

Week	Topics	Key Concepts
1-2	Functions, Pre-Calculus Review	Types of functions, domain, range
3-4	Limits, Continuity	Limit definition, continuity
5-7	Derivatives	Definition, rules, applications
8-10	Applications of Derivatives	Optimization, related rates, curve sketching
11-13	Integration	Antiderivatives, definite/indefinite integrals
14-15	Advanced Integration, Review	U-substitution, area between curves, exam prep

Additional info: This syllabus covers all major topics listed in the Calculus I curriculum, including functions, limits, derivatives, applications, and integration. It is suitable for exam preparation and provides a logical sequence for mastering calculus concepts.