MATH 1081 - Single Variable Calculus I
Hours/Week: Lecture None Lab None
Course Description: This is the first course in the two-semester sequence of Single Variable Calculus. Topics include functions of a single variable, limits and continuity, differentiation of algebraic and transcendental functions, the chain rule, anti-differentiation, Riemann sums, indefinite and definite integrals, and the Fundamental Theorem of Calculus, with associated applications in each area. A graphing calculator is required. Instruction will be provided in the use of the TI-84 calculator. This course is intended for students majoring in chemistry, engineering, physics, science, mathematics, mathematics education, and computer science.
4 Mathematics/Logical Reasoning
Prerequisite(s): Course placement into MATH 1081 or above or completion of MATH 1062 with a grade of C or higher.
Recommendation: Eligible for college-level Reading and English.
a. Intuitive understanding
b. Limit theorems
c. Calculating limits
d. Infinite limits and limits at infinity
- Intuitive understanding
- Definition of continuity at a point
- Continuity on an interval
3. The Derivative
- Intuitive or geometric interpretation
- Numerical derivatives
- Definition of the derivative
- Differentiation formulas
- Composite functions and the chain rule
- Higher order derivatives
- Implicit differentiation
- Logarithmic differentiation
- Derivatives of Parametric equations
- Derivatives of hyperbolic functions
4. Graphing Functions
- Increasing and decreasing functions
- Extreme values
- End behavior and asymptotic behavior
- First derivative test
- Second derivative test
- Curve sketching
5. Applications of the Derivative
- Tangent line equations
- Related rates
- Linear approximations
- Newton’s Method
- L’Hospital’s Rule
- Position, velocity, and acceleration
6. Anti-Differentiation and the Indefinite Integral
- Transcendental functions
- Initial value problems
7. The Definite Integral
- Intuitive understanding
- Approximating sums
- Riemann sums
- Area under curves
- The Fundamental Theorem of Calculus
At the end of this course students will be able to:
- demonstrate critical and logical reasoning when solving problems.
- explain the concept of limit from a graphical, numerical, and algebraic point of view.
- illustrate and calculate limits of a variety of algebraic and transcendental functions, and limits involving infinity.
- describe what it means for a function to be continuous and identify various types of discontinuities.
- determine the derivative of a function graphically, numerically, and symbolically.
- compute a derivative using the definition.
- find derivatives using differentiation rules, implicit differentiation, and logarithmic differentiation.
- recognize the derivative as a rate of change and as a slope of the tangent line.
- solve application problems including optimization and related rates using derivatives.
- compute derivatives of parametric equations.
- analyze important features of the graph of a function using the first and second derivative tests and limits.
- explain the relationships between a function, its derivative, and its second derivative, numerically, graphically, and symbolically.
- recognize limits in indeterminate forms (quotient, product, difference, power) and apply L’Hospital’s Rule appropriately to evaluate them.
- find the anti-derivative of fundamental algebraic and transcendental functions.
- approximate the area under a curve with approximating sums by hand and with technology.
- define the definite integral as a limit of Riemann sums.
- describe the relationship between derivative and definite integral as expressed in both parts of the Fundamental Theorem of Calculus, and apply it to evaluate definite integrals using anti-derivatives.
- communicate clearly a problem’s solution and its explanation for the intended audience in terms of the problem posed.
Competency 1 (1-6)
04. 01. Illustrate historical and contemporary applications of mathematical/logical systems.
04. 02. Clearly express mathematical/logical ideas in writing.
04. 04. Apply higher-order problem-solving and/or modeling strategies.
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