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Strongly recommended but not required: Complete the Recommended Review of Calculus I Topics
Module 1: Antiderivatives, Definite and Indefinite Integrals, Integration by Substitution, Integration by Parts
Assignments 0 - 4
Text Sections 6.1 - 7.2
1. Graphically and numerically construct approximate antiderivatives of functions given graphically, analytically or numerically.
2. Use the Fundamental Theorem to analytically determine definite and indefinite integrals of function which are given analytically.
1. Solve differential equations of the form dy/dx = f(x) with initial condition y(a) = b.
2. Construct the family of solution curves for a differential equation of the form dy/dx = f(x) with initial condition y(a) = b.
3. Apply the Second Fundamental Theorem and the chain rule to find derivatives of the form d/dt ( integral(f(x), x from a to g(t) ).
4. Apply the Second Fundamental Theorem to graphically construct integral( f(x), x from a to t)
1. From the premise of constant acceleration derive and apply the functions describing the velocity and position of a uniformly accelerating object as functions of time.
2. Apply Newton's Second Law and the definition of work to solve problems.
3. Use the method of substitution to find indefinite integrals, and check by differentiating the result.
4. Having found an indefinite integral, use it to find a specified definite integral.
1. Use the method of integration by parts to find indefinite integrals, and check by differentiating the result.
2. Having found an indefinite integral, use it to find a specified definite integral.
(Complete Test #1 on Chapter 6 and Sections 7.1-7.2)
Module 2: Integration Techniques, Indefinite Integrals, Applications of Integration
Assignments 5 - 11
Text Sections 7.3 - 8.8
1. Use a given table of integrals to find definite and indefinite integrals of given functions.
2. Apply the method of partial fractions to integrate a given rational function.
1. Apply appropriate trigonometric substitutions to integrate given expressions.
1. Explain the left- and right-hand rules for approximating definite integrals.
2. Explain the midpoint and trapezoidal rules for approximating definite integrals.
3. Explain and illustrate the nature of the errors resulting from midpoint and trapezoidal rules for functions which are concave up or concave down.
4. Explain why, when using midpoint or trapezoidal rule, doubling the number of intervals tends to result in about one-fourth the error.
5. Apply midpoint, trapezoidal and Simpson's rules.
6. Verify the expected behavior of errors for midpoint, trapezoidal and Simpson's rules.
1. Use appropriate limits to evaluate various improper integrals when one or more of the limits of integration are infinite.
2. Use appropriate limits to evaluate various improper integrals when the integrand becomes infinite.
3. Use appropriate limits to determine whether a given improper integral converges or diverges.
4. Prove for which values of p the integral of 1 / x^p, x from 1 to infinity converges and for which it diverges.
5. Prove for which values of a the integral of e^(-alpha * x), x from 0 to infinity, converges and for which it diverges.
6. Apply appropriate comparison tests to determine convergence or divergence of given integrals.
7. Use definite integrals based on horizontal and/or vertical slicing to find the volumes of given spherical sections, cones, pyramids and similar solids.
8. Use definite integrals to find volumes of solids of revolution.
9. Use definite integrals to find volumes of regions where sufficient information in know about cross-sections.
10. Use definite integrals to find the arc length of a curve given by y = f(x), a <= x < = b.
11. For any of the integrals in 7 - 10 above explain what interval of what axis was partitioned to obtain the integral, and how the integrand represents the volume or arc length corresponding to a typical subinterval of the partition.
1. Graph a given function r = f(theta) in polar coordinates.
2. Explain why the region of the polar graph of r = f(theta) corresponding to the interval from theta to theta + `dTheta is close to the area of a triangle whose base is r = f(theta) and whose altitude is r `dTheta = f(theta) `dTheta.
3. Use integration to find the area of the region enclosed by theta = alpha, theta = beta and the polar graph of r = f(theta).
4. For any the integrals in 3 above explain what interval was partitioned to obtain the integral, and how the integrand represents the area corresponding to a typical subinterval of the partition.
5. Define the moment of a system of discrete masses at known positions along an axis, relative to a given point of rotation.
6. Define the center of mass for a system of discrete masses at known positions along an axis.
7. By partitioning an appropriate interval set up the integral for the mass of an object lying along the x axis, given its density function.
8. By partitioning an appropriate interval set up the integral for the moment about a selected point of an object lying along the x axis, given its density function.
9. Find the center of mass of an object lying along the x axis, given its density function.
10. For a 3-dimensional object of constant density find the coordinates of its center of mass.
11. Apply the process of partitioning an appropriate interval to obtain the required integral to calculate the work done by a variable force exerted through an interval of position, the work required to assemble a given 3-dimensional object against the force of gravity, the force exerted by a variable pressure on a given surface.
1. Determine present and future value of a payment made at a given time, relative to the present time and some specified future time.
2. By partitioning an appropriate time interval derive the integral for the present or future value of a given income stream.
3. Given supply and demand curves, in graphical form, estimate equilibrium values of price and quantity, as well as producer and consumer surplus.
4. Given supply and demand functions, determine equilibrium values of price and quantity, as well as producer and consumer surplus.
1. Explain the relationship between a probability density function and a probability distribution function.
2. Determine whether a given function is a probability distribution function; if not determine whether multiplication by a constant would make it a probability distribution function and if this is the case determine the constant.
3. Given the graph of a probability density function sketch a reasonable approximation of the graph of the cumulative probability distribution function, and vice versa.
4. Given the graph of a probability density function or a cumulative probability distribution function, estimate the probability that an event will occur within a given interval.
5. Identify the characteristics of a probability density function or cumulative probability distribution function which are related to the specific situation modeled by that function.
6. Given a probability density function or a cumulative probability distribution function, find the other.
7. Given a probability density function, partition a given interval to set up an integral for the probability that an event will occur within that interval.
8. Given a probability density function or a cumulative probability distribution function, find the probability that an event will occur within a given interval.
9. Find the mean and median values of a random variable, given its probability distribution.
10. Find the integral for the probability that a random variable with normal distribution having given mean and standard deviation will occur within a given interval.
(Complete Test #2 on topics covered through Chapter 8)
Module 3: Sequences and Series
Assignments 12 - 16
Text Chapters 9 and 10
1. Generate elements of a sequence given an algebraic or recursive definition of the sequence.
2. Graphically represent a given sequence.
3. Know and apply the definition of convergence of a sequence.
4. Find the sum of a finite or infinite geometric series.
5. Know and apply the definition of convergence of an infinite series.
6. Know and apply the convergence properties of series.
7. Know and be able to prove the theorem for convergence or divergence of p series.
8. Apply the integral test to determine convergence or divergence of given infinite series.
1. Apply the comparison and/or limit comparison test to determine convergence or divergence of given infinite series, or show that a specified test does not apply.
2. Where applicable apply the test of absolute convergence to determine convergence or divergence of given infinite series.
3. Where applicable apply the ratio test to determine convergence or divergence of given infinite series.
4. Explain how the ratio is constitutes a comparison with geometric series.
5. Where applicable apply the alternating series test, including error bounds, to determine convergence or divergence of given infinite series.
6. Know and apply the definitions of absolute and conditional convergence.
7. Know and apply the definition of power series, and the condition for convergence of a power series.
8. Numerically and graphically represent the behavior of power series for convergent and divergent values of the variable.
9. Apply the ratio test to determine the radius and interval of convergence of a given power series.
1. Understand and be able to find linear and quadratic approximations of functions, and explain how the process is extended to higher-order approximations.
2. Construct the Taylor polynomial of given degree for a given function at a given point.
3. Verify numerically and/or graphically how the Taylor expansion increases in accuracy as the degree of the polynomial increases.
4. Construct the Taylor series for a given function at a given point.
5. Find the interval of convergence for a given Taylor series.
6. Find the Taylor series for (1 + x) ^ p.
1. Given the Taylor series of a function f(x), find the Taylor series of f(g(x)), where g(x) is known as a polynomial or Taylor series.
2. Differentiate or integrate the Taylor series.
3. Find and apply error bound for Taylor series.
4. Prove convergence of Taylor series by showing that error bounds approach zero.
1. Calculate Fourier coefficients, Fourier approximations and Fourier series of given periodic functions.
2. Calculate given harmonics, the energies of the various harmonics and the total energy of a given pulse train.
Module 4: Introduction to Differential Equations
Assignments 17 - 21
Text Chapter 11
1. Solve, algebraically or numerically, differential equations of the form dy/dx = f(x) and graphically represent the family of solutions.
2. Solve, algebraically or numerically, differential equations of the form dy/dx = f(x) with initial condition x(a) = b.
3. Know the meaning of the order of a differential equation and the implications for the number of arbitrary constants to be expected in the solution.
4. Given a first-order differential equation plot its slope field and use to construct a representative family of solution curves, and/or the specific solution curve for a given initial condition.
5. Given a slope field, match it with one of a given set of differential equations.
6. Given a slope field, determine interval(s) over which a solution to the associated equation is expected.
1. Apply Euler's Method to obtain an approximate solution to a given first-order differential equation, and assess the accuracy of the approximation.
2. Given a first-order differential equation, determine if the variables can be separated and if so use this method to solve the equation.
1. Solve, by setting up and solving a differential equation, problems involving growth, decay, Newton's Law of Cooling, and situations with similar behavior.
2. Apply the definitions of stable and unstable equilibrium to determine the nature of a given solution.
1. Set up, solve and interpret solutions in a variety of applications problems using differential equations, including the logistic equation.
2. Use the phase plane to investigate the behavior of solutions to a given system of differential equations.
1. Use nullclines to analyze the nature of the solution trajectories of a system of two simultaneous first-order differential equations, and the nature of various equilibrium solutions.
2. Apply second-order differential equations with constant coefficients to mechanical systems and oscillating circuits and interpret solutions.