Math 121 Calculus 1 Fall 2009 Outcomes List for Final Exam This outcomes list summarizes what skills and knowledge you should have reviewed and/or acquired during this entire quarter in Math 121, and what sort of problems to expect on material which was covered after exams 1, 2, and 3. The problems here are representative, although we do not guarantee that the problems on the exam will look exactly like the ones here. Most are from homework problems assigned so far and some are extra problems from the book. You should, of course, also review the problems assigned on the syllabus. The final exam will be ***comprehensive***. In addition to the material covered on exams 1, 2, and 3, it will cover 3.3, 3.4, 3.5, 3.6, 4.1, 4.2, 4.3, 4.4, and 4.5. For the new material for the final exam, you should be able to: Section 3.3: (Exponential Functions and Inverse Trig Functions) Given an invertible function f, calculate the derivative of f^{-1} without explicitly obtaining f^{-1}, by using formula (2) of this section. Use the ideas of these three sections to differentiate exponential functions and the inverse trigonometric functions. Section 3.3): The computation of the derivative of as described in the book, using the trick of figure 3.3.3 (or any alternative method you may know). Example 4 is a good example of logarithmic differentiation. Problems 30, 32. Section 3.4: (Related Rates) Be able to follow the steps listed before Example 3 of this section to solve related rates problems. Be able to read the information and write down the related quantities, and differentiate the relation (using the chain rule appropriately). Section 3.4): Example 4, Example 5; Problems 18, 20. Section 3.5: (Local Linear Approximation and differentials) Be able to use the notion of local linear approximation to give approximate values of f(a + dx), where f(x) is some given function, a and dx given numerical values, and both f(a) and f (a) are known. Section 3.5): Problems 24, 26, 30.
Section 3.6: (L Hospital s Rule) Use the techniques of differentiation encountered thus far to employ L'Hopital's rule for the evaluation of indeterminate limits. Given a limit not necessarily of the standard indeterminate form, convert it to such a form (if possible) and evaluate. Know that L Hopital s rule does NOT apply to determinant limits. Section 3.6): Problems 54, 56 Section 4.1: (Analysis of Functions) Use the first derivative to determine where a function is increasing, decreasing, or constant as x increases. Identify critical points and stationary points. Define the concavity of a function and use the second derivative to determine intervals of concavity up and down, and the location of inflection points. Section 4.1): Problems 28, 30 Section 4.2: (Relative Extrema and Graphing Functions) Define relative extrema and locate them for a given function. Relate these to the behavior of polynomials at simple or multiple roots. Combine with an understanding of the end behavior of polynomials ("dominant term") to quickly sketch graphs of polynomials. Section 4.2): Example 6, Problem 36 Section 4.3: (Rational Functions, Cusps and Vertical Tangents) Determine whether a function has cusps or vertical tangent lines, or asymptotes (vertical, horizontal, oblique, or curvilinear). Sketch rational functions using this and extremum information. Section 4.3): Example 2, Example 4, Problem 25. Section 4.4: (Absolute Max-Min) Locate the absolute maxima and minima of a function over a given interval. Section 4.4): Example 2, Problem 14 Section 4.5: (Applied Max-Min) Use the above to solve optimization problems: given a system of related quantities, one of which is to be optimized in some sense, find the set of values for the quantities that optimizes the desired value.
Section 4.5): Example 3, 4; Problems 50, 56, 58. This is the end of the review for material on the final exam which was taught after exam 3. Exam 3 will cover sections 2.3, 2.4, 2.5, 2.6, 3.1, and 3.2. Sections 3.3, 3.4, 3.5 will ** *NOT*** be covered on Exam 3. Topics: Techniques of differentiation, product and quotient rule, derivatives of trig functions, chain rule, implicit differentiation, derivatives of logarithmic functions. For this exam, you should be able to: Section 2.3: Give the derivative of x^n for any real n. Relate differentiation to real linear combinations of functions: sums, differences, and constant multiples. Define and calculate higher-order derivatives. Section 2.3): Example 8 ; Regular problems 65, 67, 69. Section 2.4: State and use the product and quotient rules for differentiation. Section 2.4): Example 4; Regular problems 29, 31, 33. Section 2.5: Give the derivative of any trigonometric function. Memorize limit formulas (1) and (2) from this section. Be able to derive these results using the limits discussed earlier, in combination with the limit rules and several trigonometric identities. Section 2.5): Example 4; Problem 34, 39. (continued on next page) Section 2.6: Use the chain rule to differentiate compositions of functions. Section 2.6): Example 4; Problems 6, 28, 35, 56, 57, 75.
dy Section: 3.1: Use implicit differentiation to determine when x and y are related dx implicitly, without explicitly solving for y. Section 3.1): Example 3, Example 4 (do it both ways!); Problems 33, 42. Section 3.2: Differentiate functions logb ( x), in particular ln(x ). Be able to use logarithmic differentiation for products and quotients. Section 3.2): Example 5; Problems 39, 40, 53, 55 This is the end of material for review of Exam 3. Exam 2 will cover sections 1.3, 1.5, 1.6, 2.1, 2.2 Chapter 1: Introduction to limits. In this chapter, we learn how to do the fundamental calculations with limits that are required for chapter 2, where it is used to define and calculate the derivative. 1.3 Be able to compute limits at infinity. Understand that the symbol does NOT mean the number 1; it indicates an ambiguous expression which has to be evaluated using the techniques described in this section. Know how to take limits of rational functions. Know how to use the rationalization trick to handle expressions with radicals. Section 1.3): Examples 10, 11; problems 61, 62 (Equation 7 of this chapter is worth memorizing) 1.5 Understand the informal meaning of continuity. Understand the intermediate value theorem. Be able to check if a function has a removable singularity or not. Be able to check continuity of a function over an interval, including checking this property at the end points of the interval. Section 1.5): Example 2, 4; problems 35, 36. 31 is a regularly assigned problem, but is particularly good at illustrating concepts from this section.
1.6 Know, and be able to use, the most important limits described in this section: the equations from theorem 1.6.5. Section 1.6): Example 4; Regular problems 54, 56; (continued on next page) 2.1 Be able to compute average rates of changes, given numerical data or an algebraic description of the functional relation of the dependent and independent variables. Be able to compute slopes of secant lines for graphs of functions. Using a limiting process, find slopes of tangent lines for graphs of functions (in section 2.2, you see that this is the same as computing the derivative). Section 2.1): Example 4, 5, 6; Regular problems 24, 26, 28 2.2 Know the fundamental definition of the derivative, in its equivalent forms. Know that the geometric interpretation of the derivative is the slope of the tangent line; and that its interpretation in physics is as the instantaneous velocity. Given the graph of a function, be able to sketch the graph of its derivative function. Know, in particular, how to recognize horizontal tangent lines, vertical tangent lines, and how that affects the graph of the derivative. Section 2.2): Example 2; Regular problems 26, 32. The material for Exam 2 ends here Exam 1 will cover sections 0.1, 0.2, 0.4, Appendix B, 0.5, 1.1, 1.2 Chapter 0: Pre-calculus fundamentals. In this chapter we quickly review the knowledge we expect students to have as we approach the rest of the course.
0.1 Define functions and determine when a relation among quantities constitutes a function, i.e., by the vertical line test. Given a function, determine its domain and range. Understand how algebraic operations affect domains and range. Section 0.1): Examples 6, 7; Problems 24, 27, 36 0.2 Define operations on functions, such as addition of functions, multiplication, and composition. Be able to find the domain of a composition. Section 0.2): Examples 7, 8; Problems 52, 54, 68 0.4 Given a function, determine whether it has an inverse and, if so, what this inverse is. Apply be able to discuss the range and domain of the function and its inverse. Relate the graph of a function and its inverse. Section 0.4): Example 6; Problems 25, 28. 0.5 Describe what is meant by exponential growth or decay, and relate these to exponential functions b^x and b^{-x}. Relate exponentiation to logarithms, especially the natural log. State and use the basic properties of exponents and logs that are useful for calculation. Section 0.5): Example 4; Problems 26, 28 Appendix B. You should know the definition of the auxiliary functions sec, cosec, tan, and cotan, in terms of sin and cos. You should know, and be able to use, the basic trig 2 2 identity sin ( x ) cos ( x) 1, as well as the sin and cos addition formulas. You should know the sin and cos of the standard angles,,,, as well as related angles in the 6 4 3 2 different quadrants.
Appendix B): Example 6; Problems 7, 13, 34. Chapter 1: The Limit. The limit is the basic object defining calculus. In this chapter we define the limit and develop techniques for finding limits of various functions. 1.1 Calculate the limit of a function f of a variable x, as x approaches some finite value c. Determine when such a limit does not exist, as a one-sided or two-sided limit. In either case, describe the behavior of f in the vicinity of such a point. Include an understanding of what is meant by increase or decrease without bound in the vicinity of such a point, or a vertical asymptote. Section 1.1): Problem 8 1.2 Discuss how the limit interacts with basic arithmetic operations such as addition, multiplication, and division. Take limits of functions involving especially quotients or radicals; define some indeterminate forms. Section 1.2): Examples 9, 10; Problems 38, 40