MTH 251 Differential Calculus

Lecture Hours: 5
Credits: 5

Prepares students for further study in mathematics, sciences, engineering and other technical areas. Covers limits, continuity, rates of change, and derivatives with applications. Introduces the indefinite integral.

Prerequisite: Placement into WR 115 (or higher), or completion of WR 090 (or higher); and placement into MTH 251 (or higher), or completion of MTH 112Z (or higher) or equivalent course as determined by instructor; or consent of instructor. (All prerequisite courses must be completed with a grade of C or better.)
Student Learning Outcomes:

Create mathematical models of abstract and real-world situations using first and higher order derivative and -derivative functions.
Use inductive reasoning to develop mathematical conjectures involving first and higher order derivative and anti-derivative function models. Use deductive reasoning to verify and apply mathematical arguments involving these models.
Use mathematical problem-solving techniques involving first and higher order derivative and anti-derivative functions, including the use of graphical, symbolic, narrative and tabular representations.
Make mathematical connections and solve problems from other disciplines involving first and higher order derivative and anti-derivative functions.
Use oral and written skills to individually and collaboratively communicate about first and higher order derivative and anti-derivative functions.
Use appropriate technology to enhance mathematical thinking and understanding, to solve mathematical problems involving first and higher order derivative and anti-derivative functions and judge the reasonableness of results.

Statewide General Education Outcomes:

Use appropriate mathematics to solve problems.
Recognize which mathematical concepts are applicable to a scenario, apply appropriate mathematics and technology in its analysis, and then accurately interpret, validate, and communicate the results.

Content Outline

Limits
- Definition of limit
- Limit laws
- Limits involving infinity
- Limits related to rates of change
- Continuity
Derivative
- Average rate of change
- Instantaneous rate of change
- Derivative function
- Interpretations of the derivative
- Second derivative
- Differentiability
- Linearization
- Antiderivatives
Rules of Differentiation
- Power and polynomial
- Exponential, logarithmic, trig, and inverse trig
- Chain
- Sum, difference, product, and quotient
- Implicit
Applications of Derivatives and Antiderivatives
- First derivative test
- Second derivative test
- Optimization
- Related rates
- Mean value theorem
- Other disciplines (e.g. Engineering, Economics, Chemistry)
- Rectilinear motion
- Roots (e.g. Newton’s method, Intermediate value theorem
- Differentials
- L’Hospital’s rule