The Final Exam will be held in our regular classroom, 241 Altgeld, from 1:30 to 4:30 pm, Wednesday, December 13. The following topics will be covered on the exam:
Chapter 6 (Know what Ln, Rn , Mn, and Tn are from Chp 5 and how to compute them)
- Section 6.1:
- Know the ways in which bounds Lnand Rn are related to increasing/decreasing functions f
- Know how bounds Mn, and Tn are connected to concave up and concave down.functions f
- Section 6.2:
- Theorem 3 is the important result here
- Know how to find values of K1 and K2 and estimate the size of n needed for a given error tolerance
Chapter 7
- Section 7.1:
- Be prepared to compute the area between two curves using either x strips or y strips
- Know the arc length formula, how it was formed, and how to use it in simple examples
- Section 7.2:
- Know how to compute volumes of solids using the slice method
- Section 7.3:
- Be familiar with the formula work = force x distance
- Know how to compute the total work done when an object is moved along a line by a force F(x)
- Know how to compute the work required to empty a tank of fluid
- Section 7.4:
- Know how to identify and solve differential equations with variables separable
- What is an Initial Value Problem and how does it help you determine the constant of integration C ?
Notes on Surface Area
- Be familiar with the formula for the surface area of a surface of revolution and how that formula is derived
- Know how to compute surface area for simple surfaces of revolution
Notes on Center of Mass
- How do we define the total mass of a linear object with a specified mass density function?
- How do we define the center of mass for a linear object with a specified mass density function?
- Be able to compute total mass and center of mass in simple examples
Chapter 8
- Section 8.1:
- Be able to do integration by parts forwards and backwards
- Know what is meant by a reduction formula and how integration by parts produces such formulas
- Section 8.2:
- Know how to express a rational function in terms of a partial fraction decomposition
- Know how to integrate rational functions using partial fraction decompositions
- Section 8.3:
- Know how to integrate expressions of the form sinnx cosmx and secnx tanmx
- Know how to integrate expressions that involve Sqrt(a2 - x2), Sqrt(a2 + x2), and Sqrt(x2 - a2)
Chapter 9
- Section 9.1:
- The main technique in this section is the computation of the nth degree Taylor Polynomial Pn of a function f about a base point x0. Make sure you know how to compute Pn.
- Know the sense in which Pn approximates f: both f and Pn and their derivatives up to order n agree at the base point x0. This is called "contact of order n at x0".
- Accuracy of using Pn in place of f near x0 should increase as n is increased.
- Section 9.2:
- Theorem 2 (Taylor's Theorem) is the fundamental result in this section. Know it forwards and backwards.
- What is a MacLaurin polynomial?
- Be able to use Theorem 2:
- Know how to find Kn+1
- Know how to estimate error on a fixed interval of x's
- Know how to find an interval that produces an error smaller than a given number.
Chapter 10
- Section 10.1:
- What makes an integral improper?
- How do we define the value of an improper integral and what do convergence and divergence mean? Know how to impliment these concepts in the case of specific improper integrals, i.e. testing convergence or divergence, and computing the integral if it is convergence.
- What do convergence and divergence mean if an integral is improper is more than one way?
- Section 10.2:
- Know the comparison theorem and how to use it in assessing convergence and divergence.
- What is the p-test for improper integrals?
- Know Theorem 2 on absolute convergence of integrals
- Know how to estimate the size of the "tail" of an improper integral, i.e. for what value of a is the improper integral from a to infinity less than a specified error?
Chapter 11
- Section 11.1:
- What does one mean by the convergence of a sequence?
- Know Theorems 1 and 2 and how to use them in practical calculations
- Be able to compute the limit of sequences, possibly using L'Hopital's rule.
- Section 11.2:
- What does one mean by the convergence and divergence and the sum of a series?
- What is a harmonic series?
- Know how to identify and compute the sum of a geometric series
- Know Theorem 5 for breaking complex series into sums of simpler series
- Know Theorem 6 - the nth term test for non-convergence
- Section 11.3:
- Know the comparison test for series and how to use it
- Know Theorem 8 on the integral test and the estimates it gives
- What is the p-test for series?
- Know how to use the ratio test to test for convergence.
- Section 11.4:
- What is meant by absolute and conditional convergence. Be able to test for these is simple problems
- Be sure to know Theorem 11 on Alternating Series, including the error estimate. You should be able to use this in simple cases to estimate the error between the sum of an alternating series and the nth partial sum
- Section 11.5:
- What is a power series, i.e. base point, coefficients?
- What does one mean by the radius of convergence of a power series?
- Know how to comput the radius of convergence of a power series using the ratio test.
- Sections 11.6 and 11.7:
- Know the table of typical power series on page 595
- Know how to generate new power series from old power series by differentiating or integrating "term by term"
- Know how to generate new power series using Taylor's formula for the coefficients in a Taylor series
- What is a MacLaurin series?
Vectors and Polar Coordinates
- Section V.1:
- Be familiar with vector valued functions and how they give rise to parametric curves
- What is the velocity vector associated with a parametric curve?
- What is the expression for the arc length of a parametric curve?
- Know how to proceed from acceleration to velocity to position for two dimentional motion with initial position and velocity prescribed
- Section V.2:
- Be able to move back and forth from cartesian points and polar points
- Be able to convert from cartesian expressions of curves to polar expressions of curves
- Section V.3:
- How does one compute slopes for points on a polar curve. Be prepared to do this in simple examples
- How how to compute the area between a polar curve and the origin.