  
Description  
 Introduction to differential calculus, including limits, derivatives and their applications to curvesketching, families of functions, and optimization.


Course Objectives  
 The student will be able to:
 Demonstrate an understanding of and compute limits of functions at real numbers.
 Determine if a function is continuous at a real number.
 Find the derivative of a function as a limit.
 Find the equation of a tangent line to a function.
 Compute derivatives using differentiation formulas.
 Use implicit differentiation.
 Graph and differentiate functions in polar and parametric form.
 Demonstrate an understanding of and calculate first, second and higherorder derivatives.
 Graph functions using methods of calculus.
 Use differentiation to solve applications such as related rate problems and optimization problems.
 Define the antiderivative and determine antiderivatives of simple functions.
 Use technology such as graphing calculators and/or computer software to assist in solving problems involving any of the topics in (A) through (K) above.
 Discuss mathematical problems and write solutions in accurate mathematical language and notation.
 Interpret mathematical solutions.

Special Facilities and/or Equipment  
  Graphing Calculator.
 When taught hybrid: Four lecture hours per week in facetoface contact and one hour per week using CCC Confer. Students need internet access.

Course Content (Body of knowledge)  
  Demonstrate an understanding of and compute limits of functions at real numbers.
 One sided and two sided limits
 Finding limits graphically
 The limit laws
 Compute limits using numerical and algebraic approaches
 Indeterminate forms and l'Hospital's Rule
 Formal definition of a limit
 Determine if a function is continuous at a real number.
 Continuity
 Intermediate Value Theorem
 Continuity and differentiability
 Find the derivative of a function as a limit.
 Use the definition to find derivatives of functions at a point.
 Use the definition to find derivatives of functions.
 Find the equation of a tangent line to a function.
 Average and instantaneous rates of change
 Slopes of secant and tangent lines
 Compute derivatives using differentiation formulas.
 Power rule
 Product rule
 Quotient rule
 Chain rule
 Differentiation of exponential functions
 Differentiation of logarithmic functions
 Differentiation of inverse functions
 Differentiation of trigonometric functions
 Differentiation of hyperbolic functions
 Use implicit differentiation.
 Equations to tangent lines at points of implicit curves
 Use implicit differentiation to find the derivative of an equation of two variables.
 Graph and differentiate functions in polar and parametric form.
 Tangents to parametric and polar curves
 Demonstrate an understanding of and calculate first, second and higherorder derivatives.
 Calculate first derivatives
 Calculate second derivatives
 Calculate higher order derivatives
 Graph functions using methods of calculus.
 Critical points
 Graphing functions using the first and second derivatives
 Relative extrema
 Global extrema
 Inflection points
 First and second derivative tests
 Second derivative and concavity
 Asymptotes
 Use differentiation to solve applications such as related rate problems and optimization problems.
 Local linearity and linear approximation
 Differentials
 Mean Value Theorem
 Related Rates
 Optimization
 Velocity and Acceleration
 Extreme Value Theorem
 Define the antiderivative and determine antiderivatives of simple functions.
 Find general antiderivatives
 Antiderivatives in the context of rectilinear motion
 Graphing antiderivatives
 Families of curves
 Use technology such as graphing calculators and/or computer software to assist in solving problems involving any of the topics in (A) through (K) above.
 Calculator/computer utilities for evaluating derivatives
 Calculator/computer utilities for constructing graphs of derivatives
 Calculator/computer utilities for estimating limits numerically
 Calculator/computer utilities for verifying solutions to optimization problems
 Discuss mathematical problems and write solutions in accurate mathematical language and notation.
 Application problems from other disciplines
 Proper notation
 Interpret mathematical solutions.
 Interpretations of the derivative.
 Explain the significance of solutions to application problems.

Methods of Evaluation  
  Written homework
 Quizzes & tests
 Proctored comprehensive final examination

Representative Text(s)  
 Stewart, James Calculus: Concepts and Contexts. 4th ed. Belmont, CA: Brooks/Cole, Cengage Learning, 2010.

Disciplines  
 Mathematics


Method of Instruction  
  Lecture
 Discussion
 Cooperative learning exercises


Lab Content  
 Not applicable.


Types and/or Examples of Required Reading, Writing and Outside of Class Assignments  
  Homework Problems: Homework problems covering subject matter from text and related material ranging from 30  60 problems per week. Students will need to employ critical thinking in order to complete assignments.
 Lecture: Five hours per week of lecture covering subject matter from text and related material. Reading and study of the textbook, related materials and notes.
 Projects: Student projects covering subject matter from textbook and related materials. Projects will require students to discuss mathematical problems,write solutions in accurate mathematical language and notation and interpret mathematical solutions. Projects may require the use of a computer algebra system such as Mathematica or MATLAB.
 Worksheets: Problems and activities covering the subject matter.
Such problems and activities will require students to think critically. Such worksheets may be completed both inside and/or outside of class.
