MATH 1081 - Single Variable Calculus I

Credits: 5
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.
MnTC Goals
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.
Corequisite(s): None
Recommendation: Eligible for college-level Reading and English.

Major Content
1.  Limits

      a. Intuitive understanding

      b. Limit theorems

     c. Calculating limits  

          i. Graphically

          ii. Numerically

          iii. Symbollically

     d. Infinite limits and limits at infinity

2.   Continuity

1. Intuitive understanding
2. Definition of continuity at a point
3. Continuity on an interval
4. Discontinuities

3.    The Derivative

1. Intuitive or geometric  interpretation
2. Numerical derivatives
3. Definition of the derivative
4. Differentiation formulas
5. Composite functions and the chain rule
6. Higher order derivatives
7. Implicit differentiation
8. Logarithmic differentiation
9. Derivatives of Parametric equations
10. Derivatives of hyperbolic functions

4.    Graphing Functions

1. Increasing and decreasing functions
2. Extreme values
3. End behavior and asymptotic behavior
4. First derivative test
5. Second derivative test
6. Concavity
7. Curve sketching

5.     Applications of the Derivative

1. Tangent line equations
2. Optimization
3. Related rates
4. Linear approximations
5. Newton’s Method
6. L’Hospital’s Rule
7. Position, velocity, and acceleration

6.     Anti-Differentiation and the Indefinite Integral

1. Polynomials
2. Transcendental functions
3. Initial value problems

7.    The Definite Integral

1. Intuitive understanding
2. Approximating sums
3. Riemann sums
4. Area under curves
5. The Fundamental Theorem of Calculus

1. demonstrate critical and logical reasoning when solving problems.
2. explain the concept of limit from a graphical, numerical, and algebraic point of view.
3. illustrate and calculate limits of a variety of algebraic and transcendental functions, and limits involving infinity.
4. describe what it means for a function to be continuous and identify various types of discontinuities.
5. determine the derivative of a function graphically, numerically, and symbolically.
6. compute a derivative using the definition.
7. find derivatives using differentiation rules, implicit differentiation, and logarithmic differentiation.
8. recognize the derivative as a rate of change and as a slope of the tangent line.
9. solve application problems including optimization and related rates using derivatives.
10. compute derivatives of parametric equations.
11. analyze important features of the graph of a function using the first and second derivative tests and limits.
12. explain the relationships between a function, its derivative, and its second derivative, numerically, graphically, and symbolically.
13. recognize limits in indeterminate forms (quotient, product, difference, power) and apply L’Hospital’s Rule appropriately to evaluate them.
14. find the anti-derivative of fundamental algebraic and transcendental functions.
15. approximate the area under a curve with approximating sums by hand and with technology.
16. define the definite integral as a limit of Riemann sums.
17. 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.
18. communicate clearly a problem’s solution and its explanation for the intended audience in terms of the problem posed.

04. 02. Clearly express mathematical/logical ideas in writing.

04. 04. Apply higher-order problem-solving and/or modeling strategies.

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