MATH 1082 - Single Variable Calculus II

Credits: 5
Hours/Week: Lecture None Lab None
Course Description: This is the second course in the two-semester sequence of Single Variable Calculus. Topics include techniques of integration, applications of the definite integral, numerical integration, improper integrals, infinite sequences and series, Taylor series representations, parametric curves, polar curves, and elementary differential equations. 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): MATH 1081  with a grade of C or higher.
Corequisite(s): None
Recommendation: Eligible for college-level Reading and English.

Major Content
1. The Definite Integral

a. Fundamental Theorem of Calculus

b. Mean Value Theorem for Integrals

c. Average value of a function

2.  Numerical Integration

        a. Left and right sums

        b. Trapezoidal rule

        c. Midpoint sums

        d. Simpson’s rule

        e. Error calculations     

3. Applications of the Definite Integral

        a. Area between curves

        b. Volume

                  i. Cross-sections

                  ii. Discs and washers

                  iii. Shells

        c. Work

        d. Arc length

        e. Surface area

        f. Fluid force

        g. Probability

        h. Center of Mass

4. Techniques of Integration

        a. Substitution

        b. Integration by parts

        c. Trigonometric integration

        d. Partial fraction decomposition

        e. Trigonometric substitution

5. Improper integrals

        a. Convergence or divergence

        b. Indeterminate forms

        c. L’Hospital’s rule

6. Infinite Series

        a. Finite and infinite sequences

        b. Sequence of partial sums

        c. Finite and infinite series

        d. General term formula

        e. Tests for convergence or divergence

i. nth term test for divergence

                 ii. Integral test

                 iii. Ratio test

                 iv. Root test

                 v. Comparison test

                 vi. Limit comparison test

                 vii. Alternating series test

        f. Absolute convergence

        g. Taylor and Maclaurin series

        h. Radius and interval of convergence

        i. Calculus and algebra of power series

7. Polar Curves and Parametric Equations

        a. Polar coordinates

        b. Polar curves

        c. Parametric equations

        d. Conversions between rectangular, parametric, and polar equations

        e. Graphing parametric and polar curves

        f. Calculus in polar coordinates

        g. Calculus of parametric curves

        h. Tangent line calculations

        i. Area bounded by a parametric curve and polar curve

        j. Arc length

8. Elementary Differential Equations

        a. Solutions of differential equations

        b. Initial value problems

        c. Slope fields

        d. Euler’s method

        e. Separation of variables

        f. Applications and modeling with differential equations

        g. Position, velocity, and acceleration
Learning Outcomes
At the end of this course students will be able to:

1. demonstrate critical and logical reasoning when solving problems.
2. determine the integral of a function graphically, numerically, and symbolically.
3. apply a variety of integration techniques, including substitution, integration by parts, trigonometric substitution, and partial fraction decomposition.
4. explain and apply the Mean Value Theorem for integrals.
5. solve problems such as finding area, work, volume, arc length, fluid forces, center of mass, and probability using definite integrals.
6. determine convergence or divergence of an improper integral.
7. approximate area with approximating sums by hand and with technology.
8. approximate a definite integral using Simpson’s Rule and the Trapezoidal Rule.
9. apply the definition of convergence to calculate the limit of a sequence or the sum of a convergent series.
10. apply tests of convergence to determine the behavior of an infinite series.
11. find Taylor series representations of basic functions, including radius and interval of convergence.
12. find the slope and equation of a line tangent to a parametric curve.
13. graph functions in polar coordinates and find slopes of tangent lines.
14. find the arc length of parametric and polar curves.
15. find the area bounded by a polar curve.
16. solve elementary differential equations and their applications, graphically, numerically, and symbolically.
17. 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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