APPLIED MATHEMATICS

Course Credits: 4 Units

Prerequisites: Math 18

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MATH 53 - Calculus I

Course Description

MATH 53 is the first in the calculus series of three courses. It focuses on limits and continuity; differentiation; derivatives of algebraic and trigonometric functions; applications of derivatives; extrema of functions; optimizations; antidifferentiation; definite integrals; and applications of integrals.

Course Learning Outcomes

After completion of the course, the student should be able to:

  1. Discuss the fundamental concepts and processes in calculus.
  2. Use theorems and techniques to find limits, derivatives, and integrals of functions.
  3. Analyze the behavior of functions in terms of their continuity, extreme points, concavity, and asymptotes.
  4. Verify basic results in calculus through various forms of mathematical reasoning.
  5. Construct appropriate mathematical models describing situational problems.
  6. Engage in problem solving using calculus concepts, models, and tools learned.
Course Outline

I. Conics and Other Plane Curves

  1. The Polar Coordinate System, Conics in Polar Coordinates
  2. Special Polar Graphs (Cardioid, Limacon, Lemniscate, Spiral, Rose)
  3. Simultaneous Polar Equations

II. Limits, Continuity, and Derivatives

  1. The Intuitive Notion and Precise Definition of the Limit of a Function
  2. Limit Theorems; Evaluation of Limits; One-Sided Limits
  3. Infinite Limits and Vertical Asymptotes
  4. Limits at Infinity and Horizontal Asymptotes
  5. Continuity of a Function at a Point and on an Interval
  6. Discontinuous Functions; Types of Discontinuity
  7. The Tangent and Velocity Problems
  8. Derivatives and Rates of Change
  9. The Derivative as a Function
  10. Higher-Order Derivatives
  11. Differentiability and Continuity

III. Differentiation Rules

  1. Derivatives of Polynomials and Exponential Functions
  2. Product and Quotient Rules
  3. Derivatives of Trigonometric Functions
  4. The Chain Rule
  5. Implicit Differentiation
  6. Derivatives of the Inverse Trigonometric Functions
  7. Derivatives of Logarithmic Functions; Logarithmic Differentiation
  8. Derivatives of Hyperbolic Functions and Inverse Hyperbolic Functions
  9. Rates of Change in Motion and Marginal Analysis
  10. Related Rates Problems
  11. Linear Approximation and Differentials

IV. Further Applications of Differentiation

  1. Indeterminate Forms and L'Hopital's Rule
  2. Maximum and Minimum Values
  3. Rolle's Theorem and Mean Value Theorem
  4. First Derivative Test; Increasing and Decreasing Functions
  5. Second Derivative Test for Extrema and Concavity of Functions
  6. Curve Sketching
  7. Optimization Problems
  8. Antiderivatives

V. Integrals

  1. Areas and Distances
  2. The Definite Integral
  3. The Fundamental Theorem of Calculus
  4. Indefinite Integrals and The Net Change Theorem
  5. The Substitution Rule
  6. Average Value of a Function
  7. Areas of Plane Regions between Curves
  8. Volume of Solids by Slicing, Disks, Washers, and Cylindrical Shells