MATH 251 - Calculus and Analytic Geometry II

4 Credit: (4 lecture, 0 lab, 0 clinical) 4 Contact Hours: [MATH 250 ]

Objectives:

1. analyze the difference between definite and indefinite integrals.
2. apply the Fundamental Theorem of Calculus to define and analyze the antiderivative of a function.
3. apply basic integration formulas which requires identifying functions that have known antiderivatives.
4. calculate definite integrals and apply them to find the net change in a function and to solve applied problems involving net change.
5. calculate the definite integral of even and odd functions.
6. use substitution to evaluate indefinite integrals and definite integrals.
7. identify and integrate functions resulting in inverse trigonometric functions.

Goal 2:  Employ integration to solve applications.

Objectives:

1. determine the area of a region between two curves by integrating with respect to the independent variable.
2. identity when the area of a region between two curves would be easier to find by integrating with respect to the dependent variable and calculate this area.
3. determine the volume of a solid by slicing the solid into cross-sections and using integration.
4. calculate the volume of a solid of revolution using the disk method and the washer method.
5. calculate the volume of a solid of revolution using the cylindrical shells method and compare this method to the other two and determine the best method for each situation.
6. determine the length of a curve.
7. find the surface area of a solid of revolution.
8. solve applications involving density.
9. calculate the work done by a variable force acting along a line the work done pumping a liquid from one height to another.
10. determine the force exerted by a liquid on a surface.
11. find the center of mass of objects distributed along a line and of a thin plate.

Goal 3:  Investigate methods of integration.

Objectives:

1. determine antiderivatives using integration by parts.
2. solve integration problems involving products and powers of sin x and cos x.
3. solve integration problems involving the square root of a sum or difference of two squares using trigonometric substitutions.
4. determine antiderivatives of rational functions using the method of partial fractions.
5. determine antiderivatives using tables.
6. approximate the value of a definite integral by using the midpoint and trapezoidal rules.
7. recognize when the midpoint and trapezoidal rules over- or underestimate the true value of an integral.
8. show graphically the location of the errors for the midpoint and trapezoidal rules
9. determine the absolute and relative error in using a numerical integration technique and discuss an estimate in the error using the error-bound formula.
10. discuss Simpon’s rule.
11. evaluate an integral over an infinite interval and over a closed interval with an infinite discontinuity within the interval.
12. determine the convergence or divergence of an improper integral using the comparison test.
13. apply the formulas for derivatives and integrals of the hyperbolic functions.

Goal 4:  Explore sequences and series

Objectives:

1. determine the convergence or divergence of a given sequence.
2. explain the meaning of the sum of an infinite series.
3. distinguish between a sequence and a series.
4. determine the sum of a geometric series.
5. evaluate a telescoping series.
6. apply the divergence and integral test to determine the convergence of a series.
7. use the comparison test, limit comparison test, ratio test, root test, and alternating series test to test different types of series for convergence where each test is applicable.
8. estimate the sum of an alternating series.
9. explain the meaning of absolute convergence and conditional convergence.

Goal 5:  Investigate power series.

Objectives:

1. determine the radius of convergence and interval of convergence of a power series.
2. use a power series to represent a function.
3. construct power series using addition, subtraction, multiplication, substitution, differentiation, and integration.
4. describe the procedure for finding a Taylor polynomial and Taylor series for a function and demonstrate the procedure.
5. explain the meaning and significance of Taylor’s theorem with remainder.
6. estimate the remainder for a Taylor series approximation of a given function.
7. recognize the Taylor series expansions of common functions and apply techniques to find the Taylor series for a function.
8. use Taylor series to evaluate nonelementary integrals.

Goal 6:  Investigate differential equations.

Objectives:

1. use the exponential growth and decay models in applications and recognize the model as a differential equation.
2. explain what is meant by a solution to a differential equation and find the general solution to a differential equation and the solution to an initial value problem.
3. identify whether a given function is a solution to a differential equation or an initial-value problem.
4. describe the process behind constructing a direction field for a differential equation.
5. draw a solution to a first order differential equation on a direction field.
6. use Euler’s Method to approximate the solution to a first-order differential equation.
7. use separation of variables to solve a differential equation and to solve applications.