MAT 1320 Study Sheet for the final exam. Format. Topics

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1 MAT 1320 Study Sheet for the final exam August 2015 Format The exam consists of 10 Multiple Choice questions worth 1 point each, and 5 Long Answer questions worth 30 points in total. Please make sure that you provide clear, well-written and mathematically coherent and precise justification of your answers. You score points by showing that you understand the theory behind the question, know what steps to take and why, and that you can correctly use mathematical notation and terminology. For example, don t just write several equations without explaining what the relation between them is (does the first follow from the second? are you plugging in a value?). Topics The exam covers all material covered in the lectures. following sections from Stewart: This corresponds roughly to the Chapter 1: all but section 1.4 Chapter 2: entirely Chapter 3: all but section 3.7 and 3.11 Chapter 4: all but section 4.6 Chapter 5: entirely Chapter 7: sections Appendix A Concepts You should know the concepts, definitions, theorems, and rules contained in the above mentioned sections, as well as the applications discussed there and in the lectures. The following are particularly important: Background material (from Appendix A): (in)equalities, sets and intervals of numbers. Functions (domain, codomains, range). Typical problem: find the range of a given function. Properties of functions (odd, even, continuous, periodic, increasing, etc.). Typical problem: for a given function, determine whether it is increasing on a given interval. 1

2 One-to-one functions, inverses. Typical problem: find the inverse of a given function, and find the domain and range of the inverse. Composition of functions. Typical problem: given two functions, find the composite. Special classes of functions and their behaviour (polynomials, rational functions, radicals, exponential functions, logarithms, trigonometric functions). Typical problem: determine what type of function has a given graph. Rules for working with exponentials and logarithms. Typical problem: simplify an expression involving exponentials and (natural) logarithms. Tangent lines: equation of a tangent line, differentiability. Typical problem: give an equation of the tangent line to a given function at a given point. Derivative of a function and differentiability, interpretation in applications as rate of change, velocity, etc. Typical problem: using the definition, find the derivative of a given function. Area under the graph of a function, Riemann Sums. Definite integral and its properties, interpretation in applications. Typical problem: approximate the area under the graph of a given function by a given number of rectangles. Or: calculate the exact area using the definition of definite integral. Fundamental Theorem of Calculus, Parts 1 and 2. Mean Value Theorem. Typical problem: using the FToC, find the area under the graph of a given function by finding an antiderivative. Or: state the FToC. Derivatives of polynomial functions, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions, and combinations of those. Typical question: find the derivative (using any method you like, not necessarily through the definition of derivative) of a given function, or find the equation of the tangent line at a given point. Rules for differentiation: product rule, quotient rule, chain rule, power rule. Implicit differentation: understand the principles behind it (chain rule), as well as when to use it. Typical problem: find the slope of a curve at a given point on the curve, or find where the curve has horizontal tangent line.

3 Logarithmic differentiation: understand the principle behind it (taking logarithms on both sides of an equation and then differentiating both sides using implicit differentiation), and when to use it. Typical question: find the derivative of x x. Related rates: know when to use this (in situations where you need to find one derivative but can t do so directly), and how to apply it (mostly for physics problems such as inflating balloons, sliding ladders, flying kites, and so on). Exponential growth and decay: know how to derive a formula from given data, and know the standard applications (population growth, radioactive decay, continuously compounded interest). Typical question: find the size of a population of bacteria after 3 days given a certain initial population and growth rate. Linear approximations: using tangent lines to approximate a function value. Typical question: use linear approximation to estimate the square root of Newton s method: understand the theory behind it, the method for finding the sequence of approximations, and the various things that may cause the procedure to fail. Typical question: find the second approximation to f(x) = 0 starting at x 1 = 3. Or: explain why starting with x 1 = 2 does not lead to a converging sequence. Extrema: definition of local and absolute min and max. Typical problem: find the absolute and local extrema of f(x) = x 3 3x 2 6x on the closed interval [4, 4]. Critical points and the first derivative test. Typical problem: find all critical points of f(x) = arctan(x 2 ) and determine whether they are a local max, min or neither. Limits. Definition of limits and of continuity of a function. Typical problem: show, using the definition of limit, that lim xa f(x) = C. Or: state the definition of lim x a f(x) =. Algebra of limits. Indeterminate forms and LHospitals Rule. Typical problem: find lim x0+ sin(x) x. Asymptotes of a function: know when and how to determine whether a given function has vertical asymptotes. Typical problem: find the vertical asymptotes of the function f(x) = x. sin(x) Horizontal asymptotes: know how to determine whether a function has horizontal asymptotes. Typical question: find the horizontal asymptote(s) of f(x) = 3x3 16x+32 6 x 3.

4 Slant asymptotes: know when to look for these and how to show that a given function has a slant asymptote. Typical question: does f(x) = 3x3 4x+1 have a slant asymptote? 2x 2 3x+1 If so, find it. If not, explain. Second derivative: using the second derivative to find out about the concavity of a function and to determine its inflection points. Typical problem: find the inflection points of f(x) = x 4 32x 3 12x. Curve sketching: know which aspects of a function to analyze (on the exam, I will tell you explicitly which of those you have to address). Optimization problems: understand how to turn a word problem into precise mathematics. Introduce variables for all relevant quantities, write equations that capture the relationships between these quantities and identify which variable is to be minimized or maximized. Then find critical points and draw the appropriate conclusion. Mean Value Theorem: be able to state the theorem and explain what it says in concrete examples. Indefinite integrals. Know the basic forms!! (See p. 495 in the book). Also know the standard techniques for rewriting functions, e.g. using trigonometric identities. Don t forget the +C at the end, and also don t forget the absolute value in 1 x dx = ln x + C. Substitution rule: make sure you know how to handle the differentials (replace dx by an expression involving du). Also make sure to avoid mixed expressions involving both the old and the new variable. Integration by parts: know the standard applications and examples, including those where you have to use IBP more than once, e.g. as in x 2 sin(x). Trigonometric integrals: know the approach to solving those. We only briefly touched upon trig substitutions, and you will not get explicit questions about this on the final. However, since it is a special case of the substitution technique, you may wish to review it to strengthen your understanding of that technique. Study Tips: Most exam questions are based on problems covered during the lectures, examples from the book or suggested exercises. After studying the theory, your best strategy is to review as many of these as possible so that you know the general approach to these kinds of problems. However, I always change at least some aspect of the problem, so that

5 simply memorizing is not going to work: you have to understand the principles underlying the solution, so that you can handle minor variations as well.