Course Outline for Mathematics 15
Applied Calculus I

Effective: Fall 2022
SLO Rev: 10/26/2021
Catalog Description:

MTH 15 - Applied Calculus I

3.00 Units

This calculus course is intended for majors in business and in certain areas of life sciences. The course covers differential calculus of algebraic, exponential, and logarithmic functions, introduction to integral calculus, and applications in business, economics, and the life and social sciences. While this is a terminal course for many programs, some may also require the second course, MTH 16.
Prerequisite: MTH 31 or MTH 20 or MTH 21 or an appropriate skill level demonstrated through the Mathematics Assessment process.
1701.00 - Mathematics, General
Letter Grade Only
Type Units Inside of Class Hours Outside of Class Hours Total Student Learning Hours
Lecture 3.00 54.00 108.00 162.00
Laboratory 0.00 18.00 0.00 18.00
Total 3.00 72.00 108.00 180.00
Measurable Objectives:
Upon completion of this course, the student should be able to:
  1. graph polynomial, rational, exponential and logarithmic functions;
  2. find limits numerically, graphically, and using limit properties;
  3. determine intervals of continuity graphically and using continuity properties;
  4. find the derivatives of polynomial, rational, exponential, and logarithmic functions;
  5. differentiate using the definition of the derivative;
  6. find the derivatives of functions involving constants, sums, differences, products, quotients, and the chain rule;
  7. use derivatives to find rates of change and tangent lines;
  8. analyze the marginal cost, profit and revenue when given the appropriate function;
  9. sketch the graph of functions using horizontal and vertical asymptotes, intercepts, and first and second derivatives to determine intervals where the function is increasing and decreasing, maximum and minimum values, intervals of concavity and points of inflection;
  10. determine maxima and minima in optimization problems using the derivative;
  11. differentiate implicitly;
  12. solve related rate problems;
  13. find definite and indefinite integrals by using the general integral formulas, integration by substitution, and other integration techniques;
  14. find total change given rate of change;
  15. use calculus to analyze revenue, cost, and profit;
  16. use integration in business and economics applications.
Course Content:
  1. Functions and their graphs
    1. Linear functions and equations of lines
    2. Quadratic functions
    3. Polynomial functions
    4. Rational functions
    5. Exponential and logarithmic functions
  2. Limits
    1. Definitions 
    2. One-sided
    3. Infinite
  3. Continuity
    1. Definition at a point and over an interval
    2. Properties of continuity
    3. Discontinuity
    4. One-sided
  4. Derivative
    1. Limit definition
    2. Interpretation
      1. Geometric: slope of tangent line
      2. Numeric: instantaneous rate of change
    3. Rules of differentiation
      1. Constant rule
      2. Constant product rule
      3. Sum rule
      4. Product rule
      5. Quotient rule
      6. Chain rule
    4. Second derivatives
    5. Implicit differentiation
  5. Application of the derivative
    1. Maximum-minimum problems
    2. Curve sketching
    3. Related rates
    4. Approximation by increments
      1. Linear approximation of a function near point of tangency
      2. Marginal analysis
  6. Integration
    1. Antiderivatives
    2. Indefinite integrals
    3. Techniques
      1. Substitution
      2. Integration by parts
      3. Using table of integrals
    4. Definite integral
      1. Definition
      2. Evaluation
      3. Approximation by Riemann sum
    5. The Fundamental Theorem of Calculus
    6. Area between curves
    7. Applications in business and economics
Methods of Instruction:
  1. Lecture/Discussion
  2. Problem Solving
  3. Distance Education
  4. Group Activities
Assignments and Methods of Evaluating Student Progress:
  1. A company manufactures and sells x television sets per month. The monthly cost and price-demand equations are C(x) = 72,000 +60x, p= 200 + x/30, 0 < x < 6,000 1) Find the maximum revenue. 2) Find the maximum profit, the production level that will realize the maximum profit, and the price the company should charge for each television set.
  2. An apple orchard has an average yield of 36 bushels of apples/tree if tree density is 22 trees/acre. For each unit increase in tree density, the yield decreases by 2 bushels. How many trees should be planted in order to maximize the yield?
  3. It is estimated that a newly discovered oil field will produce oil at the rate of r(t)= ? thousand barrels/year, t years after production begins. Find the amount of oil that the field can be expected to yield during the first 5 years of production.
  1. Homework
  2. Quizzes
  3. Exams/Tests
  4. Final Examination
  5. Class Work
Upon the completion of this course, the student should be able to:
  1. critically analyze mathematical problems using a logical methodology;
  2. communicate mathematical ideas, understand definitions, and interpret concepts;
  3. increase confidence in understanding mathematical concepts, communicating ideas and thinking analytically.
Textbooks (Typical):
  1. Barnett, Ziegler (2019). Calculus for Business, Economics, Life Sciences, and Social Sciences (14th). Pearson.
  • Scientific or graphing calculator
Abbreviated Class Schedule Description:
Differential calculus of algebraic, exponential, and logarithmic functions; introduction to integral calculus. Applications in business, economics and the life and social sciences.
Prerequisite: MTH 31 or MTH 20 or MTH 21 or an appropriate skill level demonstrated through the Mathematics Assessment process.
Discipline:
Mathematics*