BE SURE THAT YOU HAVE LOOKED AT, THOUGHT ABOUT AND TRIED THE SUGGESTED PROBLEMS ON THIS REVIEW GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!!

Transcription

1 Review Guide for MAT0 Final Eam Part I. Thursday December 7 th during regular class time Part is worth 50% of your Final Eam grade. Syllabus approved calculators can be used on this part of the eam but are not necessary. All work will be done on the test itself; you may NOT use any scratch paper. Partial credit WILL be awarded for partially correct work so be sure to show ALL of your steps. Correct answers without the correct corresponding work are worth nothing. Questions some with parts. Since you are about to finish up this calculus class (hopefully with a passing grade) you should be able to calculate how much time that gives you per problem. Some problems will take MUCH less time than this number whereas some problems may take slightly longer. Things you should make sure that you can do! Note: Section numbers have been provided by each topic so that you can go back through your NOTES, HOMEWORK and OLD TESTS to find problems to practice. You can also go back to the class HELP page and view some of the relevant supplemental readings and videos. I have provided a few eamples for particular problems for you to practice (you should still find others of those types to practice on your own!). For those that I did not provide eamples for you should have no problem finding eamples in your notes, HW and on old tests! BE SURE THAT YOU HAVE LOOKED AT, THOUGHT ABOUT AND TRIED THE SUGGESTED PROBLEMS ON THIS REVIEW GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!!. Be able to find the absolute ma and absolute min of a given continuous function on a closed interval. Etreme Value Theorem (section.7). Like Test problem #. Be sure that you know how to use your calculator to MOST efficiently evaluate functions!. Be able to find critical numbers (values that make the derivative zero or undefined) and be able to determine open intervals on which a function is increasing and decreasing by creating the appropriate table. Be able to use the first derivative test to determine values where a function has a relative maimum and a relative minimum (section.9). Like Test problems #, 5. Be able to find possible points of inflection (the -values anyway) for a function and be able to determine open intervals on which a function is concave up and concave down by creating the appropriate table. Also be able to tell which possible points of inflection are actually points of inflection (section.0). Like Test problem # 7 (Ecept YOU may have to create the table). Note: If after practicing # if you have any questions be sure to ask in class!!!

2 . Know when L Hopital s Rule applies and how to use it to evaluate limits where direct substitution yields the appropriate indeterminate form (HW section. #, ). Like Test problem # 0 BUT be sure to also go back in your notes and HW and practice some non-polynomial fraction type eamples! For eample.. Lim or Lim 0 sin Ln 0 0 Lim direct substitution yeilds 0 sin sin Lim Lim 0 0 sin 0 cos cos 0 thus L'Hopital's rule applies. 0 0 Lim direct substitution yeilds = thus L'Hopital's rule applies Ln Ln 0 0 Lim Lim Lim Ln Remember if after applying L Hopital s rule you STILL get 0/0 then DO IT AGAIN!

3 5. In section. we did a summary of curve sketching where we put together all of the material we had learned from the previous few sections and applied it to graphing a function. Try the following problem in addition to reviewing what we did in class for section. and what you did on HW! Also see Test #5 Problems, 0 Could you go backwards? If given the graph can you fill out the table, determine domain and range in interval notation, figure out where the derivative would = 0 or be undefined, figure out the coordinates of the ACTUAL P.O.I etc? Sketch the graph of y f on the aes below given the following information!!!! The Domain of y f is 0, d The Range of y f is, a b f 0 f a 0 f f b f c f d The critical numbers (places where f = 0 or f = undefined) are a, b, c a b The "possible" points of inflection (places where f = 0 or f = undefined) occur when a,, c TEST # a a b a b a b a b a b b c c d 0, a a,, b b, c c, d Sign of f + + Sign of f a b c d This problem may look long and complicated BUT it isn t. Just plot the given points, a b f 0 f a 0 f f b f c f d and then use the information in the table to sketch the graph using the appropriate increasing/decreasing information along with sketching the appropriate curvature of the segments! Ask in class if you are unsure!

4 6. Be able to evaluate an indefinite integral (section.). Like Test 5 problems # 6 (which were ALL problems very much like YOU did in class!) 7. One day during the semester you were given the graph of a velocity vs time function and asked various questions about it. This test question will ask you four questions similar to some of those. See below for a second problem that you could use as practice (although YOUR test problem will only have parts). See the last page of this review guide for another problem with MANY parts! The graph given below represents the velocity of an object moving right and left along straight line for 6 seconds. Velocity (meters / second) m/s m/s - m/s - m/s time (seconds) A) (5 pts) Find 0 v() t dt Just find the net signed area on the interval from 0 to! = 6 B) (5 pts) What does your answer to A) represent? As we discussed in class, the net signed area represents the displacement of the object. Thus your answer of 6 in part A) means the object is 6 meters to the right of wherever it started after the first seconds. C) (5pts) What direction is the object moving from seconds to 5 seconds? Eplain HOW you know! The object is moving right during this time interval. We know this because the velocity is positive (graph is above the time ais) D) (5 pts) What direction is the object moving from 9 seconds to seconds? Eplain HOW you know! The object is moving left during this time interval. We know this because the velocity is negative (graph is below the time ais) E) (5 pts) Where is the object located at the end of its 6 second trip relative to its starting position? To answer this just calculate the net signed area over the whole trip =. Thus after 6 seconds this object is located meters to the right of wherever it started at. F) (5 pts) What is the total distance travelled by the object during its 6 second trip? To figure this out we have to calculate the actual area = Thus the object travels a total distance of 6 meters during its 6 second trip. G) (5pts) What is the object doing during the time interval,? The object is standing still during this time interval (velocity is zero). H) (5 pts) During what time interval(s) is the object moving but not accelerating? Since acceleration is the derivative of velocity the slope of the tangent line indicates the acceleration. In order to NOT be accelerating the object must have zero acceleration (or a horizontal tangent line). This occurs in and two places; 0,,. The object is not moving during the later time interval so our answer to this question is from 0 seconds to seconds.

5 8. Be able to evaluate a definite integral by using the limit definition (section.). You should be able to find MANY problems in your notes and HW to practice! Also see Test 6 # (although on your final eam you will NOT be given the limit definition so be sure that you KNOW it). Here is another problem that you could use as practice d using the limit definition of the definite integral. Use a regular partition and choose Find to be the right endpoint. * X k d b a * k k a k k n n n n n f * k * k 8 n n n k k k k (did you FOIL properly?) n n n d Lim f Lim Lim k n n n k k n n n k n 8 n n (n ) Lim 8n Lim n n 6 n n n n Be sure to practice several problems and be sure to be familiar with ALL of the summation formulas that we worked with n n n n k k k c. k k k k n k 8 9. Be able to evaluate a definite integral using the FTC (see YOUR notes and HW from section.). See Test 6 # d and see if you get the same answer as you did when you practiced the limit definition of Try derivative! d ] 6 6 See I got the same answer here as when I used the limit definition!

6 0. Be able to find the average value of a function on a given interval. Also be able to find the value(s) on that given interval that generate the average value (section.). See homework problems # 8, 9 and 0 in section.. See Test 6 #. You could also practice the following Find the average value of on, Average Value = 6 8 d Also be able to find the answer to * What is the value of on, 8. Note: This is the function and interval from question 9. that gives you the average value? Solve this one doesn't factor so solve using the quadratic formula and obtain only is in the given interval. Note: YOUR actual final eam questions #8 0 have been created to be much quicker and easier than these samples!. Be sure to review the second part of the Fundamental Theorem of Calculus (section.). For eample, can you do the following problem? Like Test 6 Problems # 6A or 6B (is one way easier to do?) d Find d t t dt. Write your answer as a polynomial in standard form with terms in descending order. d d t t dt Be able to evaluate a variety of integrals (both indefinite and definite). Some may require substitution, some may lead to natural logarithms and some may lead to inverse trigonometric function (sections.,.,.). See Test #6 ( # 8 - )as eamples! Also consider the following d 6 9 d 6 d let u = 6 d 6 du = d du d du d ln u c ln 6 c 6 u

7 NOTE: Have YOU ever seen this problem before? d d note: a = u = du d du d 9 du u d arctan arctan c c u 6 6 d u du d du d 6 d u du u It is unlikely that you will finish this test in the given amount of time unless you are EXCEPTIONALLY well prepared. You have only 65 minutes to complete as much as you can. This test may prove to be very challenging unless you have taken the necessary steps throughout the semester to learn all of the material we covered. If you haven t figured it out yet, there is no rule that says these problems must be done in the order that they appear on the test. A wise student would have prepared for this test so well by utilizing this study guide that they know eactly what questions are going to be easiest for them and complete those problems first!

8 Here is another problem like #7 with MANY different questions! vt meters/sec time (seconds) - Note: Each marking on the time ais represents one () second. An object travelling back and forth along a straight line has a velocity function given by the graph above. Answer the following questions. A. Find 0 v t dt B. What does your answer to part A. represent? represents the displacement of the object during the first two seconds. The object ends up meters to the left of wherever it started after two seconds. C. The slope of the tangent line to the graph on the time interval 0, is zero. What does that mean is happening during this two second time frame? Zero acceleration (constant velocity) D. When (give answers in the form t, t etc.) is the object speeding up, slowing down, neither speeding up nor slowing down? Speeding up =, ; 5,6 ; 7,8 Slowing down =,5 Neither =, ; 6,7 ; 8,9 E. At what time(s) does the object change direction? At t = 5 seconds (changes from going left to going right) F. What is the slope of the secant line for this graph and what does it represent? acceleration of the object during the trip.. It represents the average 9 G. Are there any point in time when the slop of the tangent line matches the slope of the secant line? Should there be? No. No not necessarily because the hypotheses of the MVT for derivatives are not all satisfied therefore we are not guaranteed that there is a point in time where the instantaneous acceleration matches the average acceleration. H. What is the total distance travelled by the object during the 9 second trip? 5m

9 I. WHERE is the object located at the end of the 9 second trip? (Give and answer like meters to the left of where it started). meters to the right of wherever it started. J. WHEN does the object pass back through the location that it started this trip? At the 8 second mark. K. What is the average value of v t on 0,9? What does this represent AND WHEN does it occur? 9 vtdt m / s 9 0. This represents the average velocity of the object during the 9 second trip. It occurs at the 5 9 second mark. Remember the /9 m/s means that if the object had just decided to set the cruise control to /9 m/s and headed to the right (since the /9 is positive) for 9 seconds then it would have ended up at the same final position as it did by travelling in the manner described in the graph.