Calculus I: Core Topics and Learning Objectives

Core Course Topics

1. Definition of Limits

The concept of limits is foundational in calculus, providing the basis for continuity, derivatives, and integrals. Students will learn to define and evaluate limits using numerical, algebraic, and graphical methods.

• Numerical Methods: Approximating limits using tables of values.

• Algebraic Methods: Simplifying expressions to find limits analytically.

• Graphical Methods: Interpreting limits from graphs of functions.

• One-Sided Limits: Understanding left-hand and right-hand limits.

• Infinite Limits: Evaluating limits that approach infinity or negative infinity.

2. Continuity

Continuity describes the behavior of functions without breaks or jumps. It is essential for understanding differentiability and integration.

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

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

• Applications: Intermediate Value Theorem and its use in root-finding.

3. Definition of the Derivative

The derivative measures the instantaneous rate of change of a function. It is defined as the limit of the average rate of change as the interval approaches zero.

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

• Interpretation: Slope of the tangent line to the curve at a point.

• Physical Meaning: Velocity as the derivative of position with respect to time.

4. Applications of the Derivative

Derivatives are used to analyze and solve real-world problems involving rates of change, optimization, and motion.

• Related Rates: Finding the rate at which one quantity changes with respect to another.

• Optimization: Maximizing or minimizing functions in applied contexts.

• Graph Analysis: Using the first and second derivatives to determine intervals of increase/decrease, concavity, and points of inflection.

5. Integration

Integration is the process of finding the area under a curve. It is the inverse operation of differentiation.

• Antiderivatives: Functions whose derivative is the given function.

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

• Fundamental Theorem of Calculus: Connects differentiation and integration.

6. Applications of Integration

Integration is used to solve problems involving area, volume, and accumulation of quantities.

• Area Between Curves: Calculating the area enclosed by two or more functions.

• Volume: Finding the volume of solids of revolution using the disk and washer methods.

• Other Applications: Work, average value, and other physical applications.

Core Course Learning Objectives

Upon completion of this course, students should be able to:

• Define and evaluate limits using numerical, algebraic, and graphical methods.

• Determine continuity and identify discontinuities in functions.

• Apply the definition of the derivative to compute derivatives from first principles.

• Use differentiation rules (product, quotient, chain) to find derivatives of algebraic, trigonometric, exponential, and logarithmic functions.

• Analyze the behavior of functions using first and second derivatives (increasing/decreasing, concavity, extrema, inflection points).

• Solve applied problems involving rates of change and optimization.

• Compute antiderivatives and evaluate definite integrals using the Fundamental Theorem of Calculus.

• Apply integration to find areas, volumes, and solve other application problems.