Review for Final Exam, MATH 170-002, Fall 2010 The test will be on Wednesday December 15 in ILC 404 (usual class room), 8:00 a.m - 10:00 a.m. Please bring a non-graphing calculator for the test. No other tools like cell-phones, computers, books or notes will be allowed on the test. The test will be comprehensive. The test will consist of 8 problems, six problems counting each 40 points, and two counting each 30 points. Please review the following topics for the Final Exam (references to Whitman Calculus): 1. Calculate limits of functions using algebraic methods, squeeze theorem, l Hospital s rule. Plug in suitable numbers to make sure that you are right. Use your calculator in a reasonable way. Is this a /, /0, 0/0 or 0/ -situation? Make clear to yourself what follows in each case! In a 0/0 or / -situation how do I proceed? Recall some algebraic tricks for situations. [2.2, 2.3, 4.3, 4.4, 4.8, 5.5] 2. Decide whether a function is continuous at some point in its domain or on some interval (recall Scotty from Star Trek). The function can be given by an expression or by a graph. [2.5] 3. Give the difference quotient whose limit is the derivative of a function at some point, calculate the derivative at some point from the difference quotient. Interpret the value of the derivative of a function at some point in terms of the function. [Chapter 2] 4. Calculate the derivatives of functions using the derivatives for special functions and the rules (including implicit differentiation and logarithmic differentiation). Simplify functions and derivatives. When you take derivatives make sure: With respect to which variable am I differentiating? Which symbols in the expression depend on this variable? Which symbols are constants?. Carefully review the applications of the chain rule (the turkey-stuffing rule): (outer(inner)) =outer (inner) times inner! Can I apply this rule in a situation f(g(h(x)))? Where is the minus sign is the quotient rule? (top/bottom) =(top bottom - top bottom )/bottom square. [Chapters 3 and 4 including 4.11, but also Chapter 6] 5. Find the equations of tangent lines at points on the graphs of functions. Sketch those tangent lines. Don t forget the order of your operations: When do I put the coordinates of the point into the equation? [Chapters 3 and 4] 6. Sketch the graph of a function following the Guidelines for curve sketching distributed in class. Do your sketch carefully! Show all intercepts, relative extrema and inflection points, the intervals where the function is increasing/decreasing, concavity, asymptotes, asymptotic behavior. What
is special about the values of the functions? Does the function have symmetry that I can use (even/odd)? (You do not have to recall the guidelines, instructions will be given!) [5.5] 7. Find the absolute maximum and minimum of a continuous function on some interval [a, b]. Find the critical points [Chapter 5, 6.1] 8. Solve an optimization problem. (see hints below) [6.2] 9. Solve a related rates problem. (see hints below)[6.3] 10. Apply Newton s method to find solutions of equations (including the calculation of roots, logarithms etc. by transferring the problem into this form). [6.3] Also recall the method from 2.5 to find roots of equations using the Intermediate Value Theorem. 11. Approximate values of functions using linear approximation by the tangent line, calculate differentials and differences of function values near a given point. [6.4] The functions could be given by a word problem (see [6.4, 4] or supplementary problems 9-13 in Schaum s Outline Chapter 21). Don t stick with the variable names, be flexible to change to other situations. (Try to understand the method: See the graph of a function and the tangent line close to the point. Do not try to recall the formula without understanding its meaning. 12. State and explain the Mean value theorem, confirm its assertion in examples, recognize why its assertion does not hold if assumptions are not satisfied. [6.5] 13. Find anti-derivatives (calculate indefinite integrals). Don t forget the constants and its geometric meaning. Why is there a constant at all? [6.5, 7.2 and 8.1] 14. Find a position of an object given an initial position and the velocity (speed) function using the anti-derivative [7.1], find the position given the acceleration, initial position and velocity, in particular for objects falling under gravitation. Describe the motion giving the velocity or position function. [9.2] 15. Approximate the position of an object using the methods in Chapter 7. Approximate the area under a curve between two given points using the methods in Chapter 7. Interpret integrals or derivatives of integrals. [7.1-7.3] 16. Compute the values of definite integrals using FTC II. [7.2] 17. Find derivatives of functions of the form G(x) = h(x) f(t)dt using FTC a I and chain rule. [7.2]
18. Find the area enclosed by the graphs of two functions or between the graph and the x-axis (over some interval). Calculate net-areas and total areas. Apply symmetry to simplify calculations. Recall that sometimes it is easier to switch from the xy-world to the yx-world. [7.3 and 9.1] 19. Calculate net-area and area, net distance and distance. Make the differences between net and total clear to yourself. [7.1, 7.2] 20. Graph derivatives and anti-derivatives of functions given the graph of the function. [Chapters 2 and 7] Some remarks: Preparation and General Hints: The problems on the test will be similar to homework problems previously assigned, problems discussed in class, problems on quizzes or exams. There will be emphasis on the last 2/3 of the class but be prepared for basic questions from the first 1/3 too. Do as many problems as you can to prepare for the test. Also work carefully through the text Whitman Calculus again. Check all examples discussed in the text. Use the additional problems in Schaum s Outline (see homework page for suggestions). Do the problems from the back side of the Graphing Guidelines. Do the problems from the practice exam before we will go through the answers on Friday. Prepare for the review days. If you have questions or problems that you are not sure about ask in class or see me in my office hours. You can also send email and I will try to answer your questions. Make a list of formulas for yourself that you think are important (derivatives, trigonometric identities etc). Go through the classes of functions (polynomial, rational, exponential, trigonometric, hyperbolic and their inverses, piecewise functions, functions containing absolute values) and make sure that you understand the basic features of each class. Do I know the derivatives of those functions, or how to find them? Recall that each derivative also gives you an anti-derivative! Review how you get graphs of shifted functions. Try to memorize the procedures and not the examples. What is the basic idea that makes this procedure work? Be aware, if you understand why the procedure works you will recall the procedure! Make clear to yourself the different notations used for derivatives, e. g. ẏ = dy dt, y = dy dx etc. What are these symbols standing for? If you are unsure about a sign have your strategies to find an answer: Like d sin(x) dx = ±cos(x)? Sketch the graph of sin(x) and use the geometric meaning of the derivative to decide on the sign. The nonzero slope at some point will suffice! Of course you will have to remember the graphs for this! Well, you will have to remember the graphs of the basic functions! Recall special situations where you have to distinguish what to do: If you apply l Hospital s rule you take top /bottom, so you do not take the derivative of top/bottom. Also be aware when you can apply l Hospital s rule and when you cannot apply it. The test can be done within a standard 50 minutes time period. Use the additional time available to carefully check all your answers. Explain all steps in
your solutions. You will not get credit for giving the correct answer for some problem without explanation. Take care when you sketch graphs. Mark the axes according to the variables, indicate the scalings. Explain all your notation. Simplify expressions for functions before you take derivatives (recall all your skills from the Precalculus times). Recall some basic principles like: Use the power rule if you can. In most cases it is easier than the quotient rule or product rule. There are some exceptions though for this principle depending on the situation like if you have to find the zeroes of the derivatives afterwards. In any case, recall to cancel in the second derivatives of rational functions before you start searching for the zeroes.) Recall your Precalculus: A fraction is zero when the numerator is zero (in case the denominator is non-zero), a product is zero when a factor is zero! Don t multiply out a factored expression when you have to find roots! Use equality signs. If you do not then I will have to be more strict with respect to errors because you are avoiding the check of the path to your result. Mathematics is not finding the right number but communicating how you found it! State your results with an equality sign or in a sentence. If you end up with a number or function in a box then you will lose points if the problem asked for more than one result. If you end up with a nonsense answer, like an area is 50 square feet or a length equal 5 cm, go back and recheck what you are doing. The same applies to curve sketching. Are the results that I get consistent with my calculations? If you think that they are not and you can t find the error, tell me that you recognize that something is wrong. You will get good credit for this. If you do not make clear that you see it doesn t work I have to assume that you don t see it! If the problem asks you to use a special method you have to show how to use the method to find the answer, even though there might be easier ways! Don t think that I won t notice those points and you will get away with it. :) Concerning the word problems: Carefully read the text and identify the key words: Write up what is asked for (Is this an optimization problem, related rates problem, differentials problems?). Draw a picture if the problem involves geometry. Indicate lengths that are fixed and lengths that change in time. If velocities are involved, what distance is changing giving the velocity (find the rest point). Sometimes you may want to sketch a picture for a particular point in time to calculate values of functions at a particular point in time, and separately the general picture for all points in time! Don t mix the two things up! If the problem asks: How fast is a distance decreasing, the answer will be a positive number, even though the derivative of the distance you are considering will have negative derivative. Think about signs! For optimization problems: If several related variables appear, which one should be eliminated to get the easiest function. What is the domain? Are the units okay? If the function involves roots, is it easier to optimize the square of the function? After you find the zeroes of the derivative: Which zero is in the allowed domain? Why is this zero a maximum/minimum (check the second derivative or argue otherwise!). Recall that sometimes an absolute extremum occus at the endpoint of the interval of interest. For related rates problems: What is the basic relation for all times
relating the invariants (there are mostly two, but also often more variables involved). When you do the differentiation recall it is implicit by time, so apply your implicit differentiation rules accordingly. Recall how ẋx appears often in formulas but that it is not true that dx dt = 2ẋx (why?). More general remarks: Do not try to recall the examples, try to recall the general set-ups. Be prepared for modifications of what is given and what is asked for. Carefully check units in application problems and distance/velocity problems! Don t forget the brackets! (If you have any questions about units ask me!) Concerning numerical results. Don t use your calculator early in the calculation. Keep fractions and constants until you have a final formula. The exception are related rates problems, where it is often easier to put the numbers into the formula after having found the relation between the rates you are comparing. Here it is not often the best to solve first for the rate we are searching for. There will only be given minimal credit on final arithmetic. Have in mind that the time you have available on the final is shorter than the time you had on the previous three exams. Read through all problems before you start working. Which ones are you feeling comfortable with, which ones not so much. Start with those you feel comfortable with. Don t overlook additional questions, there will be given credit for those too! If you get stuck ask me, there is a possibility that I can give you direction depending on how meaningful your question is. Good Luck and Happy Holidays!