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AMS 161, Applied Calculus II

Catalog Description: Analytic and numerical methods of integration; interpretations and applications of integration; differential equations models and elementary solution techniques; phase planes; Taylor series and Fourier series. Intended for CEAS majors. Not for credit in addition to MAT 127 or 132.

PrerequisitesAMS 151 or MAT 131 or MAT 126.

3 credits

FALL 2024 Course Materials - Includes eBook and access (PLEASE ATTEND THE FIRST DAY OF CLASS FOR CLARIFICATION OF COURSE MATERIALS

WebAssign for Stewart/Kokoska's "Calculus: Concepts & Contexts", 5e Single-Term Instant Access 9780357748978

 

Topics

1. Concepts on Integration and Methods of Integration: substitution, integration by parts, volume problems, approximating integrals with Riemann sums, improper integrals  - 10 hours
2. Applications of the Integral: volume and other geometric applications, parametric curves, arc lengths; probability; economic interpretations - 6 hours
3. Elements of Differential Equations: slope fields, Euler's method, applications and modeling - 7 hours
4. Systems of first-order differential equations and second-order differential equations, including solutions involving complex numbers - 8 hours.
5. A pproximations and series: Taylor series, Fourier polynomials - 5 hours
6. Review and Tests - 6 hours


Learning Outcomes for AMS 161, Applied Calculus II

1.) Demonstrate a conceptual understanding of the Fundamental Theorem of Calculus and its technical application to evaluate definite and indefinite integrals.  
        * Solve problems graphically and analytically that illustrate how integration and differentiation are inverse operations;
        * Use the Fundamental Theorem of Calculus to evaluate definite integrals whose limits are functions of x.

2.) Demonstrate skill in integrating basic mathematical functions, such as:
        * polynomials, 
        * exponential functions
        * sine and cosine functions.

3.) Develop facility with important integration tools such as:
        * reverse chain rule;
        * substitution methods; 
        * integration by parts;
        * tables of integrals.

4.) Solve problems involving geometric applications of integration: 
        * area problems;
        * volume problems;
        * arclength problems

5.) Develop basic skills with using numerical methods to evaluate integrals
       * right-hand, left-hand, and trapezoidal rules;
       * Simpson’s rule.

6.) Solve problems involving applications of integration to in physics and economics.
       * center of mass problems;
       * force problems;
       * work problems;
       * present value of multi-year investments.

7.) Solve problems with sequences and series, including:
       * find limits of sequences;
       * test series for convergence;
       * sum series.

8.) Demonstrate facility with constructing and using Taylor and Fourier series.
       * Taylor series for simple functions
       * Taylor series for composite functions and products of functions;
       * Taylor series to integration problems;
       * simple Fourier series.  

9.) Model problems with simple types of differential equations and solve these problems:
       * model problems with solve first-order linear differential equations and solve them;
       * use separation of variables to solve rate problems such as Newton’s law of cooling and logistic equations;
       * solve second-order linear differential equations.