Review Guide for MAT220Final Exam Part II.
Part 2 is worth 50% of your Final Exam grade. NO CALCULATORS are allowed on this portion of the Final Exam.
You may NOT use your own scratch paper for this Multiple Choice Exam. Instead, your instructor will give you two sheets of scratch paper that you can use to do whatever work you feel that you need to in order to obtain the answer to each question. This portion of the Final Exam consists of five pages. Each page has a blank side that can also be used for scratch work if needed. You will turn in BOTH sheets of scratch paper with your Final Exam (even if they have nothing written on them). NO PARTIAL CREDIT will be given on this portion of the Final Exam. You will have 110 minutes to complete this portion of the Final Exam (assuming that you show up on time). There are 50 questions on this part of your final exam. MANY of these questions will be very quick and require little to no written work!
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. Some problems are not necessarily specific to a section, rather will test how well you understand an idea that is covered over possibly multiple sections.
BE SURE THAT YOU HAVE LOOKED AT, THOUGHT ABOUT AND TRIED THE SUGGESTED PROBLEMS ON THIS REVIEW GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!!
1 - 2. Make sure you understand a few basic properties of antiderivatives. For example, is an antiderivative of a difference of two functions, , the same as an antiderivative of minus an antiderivative of ?(section 3.1)
True. . Does this property hold for addition and multiplication?
3 - 4. Be sure that you understand the Fundamental Theorem of Calculus (part I) and what MUST be true about , in order for to exist. If does exist, what does the answer “look like”? What does the answer to look like? (sections 3.1, 3.4)
One of these is a “family of functions” and the other is a “number”. Which is which?
5 - 6 If, does (True)? What about the converse of that statement(False)? Ask yourself this same question (both ways) about indefinite and definite integrals. i.e If does ? (True)
If does ?(False) Now answer both questions if the integral was indefinite rather than definite(True) (True).
Try to gain some insight into what each of these statements are saying!
7. If the derivative of at is zero what does that mean happens at in the graph of ? (sections 2.9, 2.12). Think carefully!
It means that the graph has a horizontal tangent line at x = c (does this mean that the graph HAS to have a relative max or min at x = c?).
8. If the second derivative of at is zero what does that mean happens at in the graph of ? (sections 2.10, 2.12). Think carefully!
It means that there is a POSSIBLE point of inflection with an x coordinate of p (i.e. it is POSSIBLE that the graph changes concavity at this location).
9 - 10. Be sure that you understand the second part of the Fundamental Theorem of Calculus (section 3.4). For example can you find Can you find for both of these examples?
11. What is the maximum number of vertical asymptotes that a function could have? What does a horizontal asymptotes represent in Calculus (how many of those could a function have)? (section 2.11, 2.12)
A function could have unlimited vertical asymptotes (y=tanx). A horizontal asymptote for
12. When doing related rates problems be sure to understand how to write down the “given” rate using derivative notation. For example if the radius of a circle is getting smaller by 2cm each second then the given rate would be
. (section 2.5)
13. Be able to find an indefinite integral (section 3.1). Example: Find
14. If an object undergoing rectilinear motion is moving right and slowing down, what would the graph of its position function look like (i.e increasing concave up, decreasing concave down etc.)? (section 2.9)
Moving right means the velocity is positive (thus the slope of the tangent line to the position graph would be positive and consequently the position graph must be increasing). If the object is ALSO slowing down then the acceleration must have the opposite sign of velocity so in THIS example that would mean negative. If the acceleration is negative the graph of the position function must be concave DOWN as acceleration is the second derivative of position. So in THIS example the graph of the position function would be increasing concave down. Can you figure out what the graph would look like if the object was moving left and slowing down, moving left and speeding up and moving right and speeding up?
15. How would you find the second derivative of a function? How would you find the “second antiderivative” of a function?
To find the second derivative of a function just take the derivative of the derivative. To find the second antiderivative just take the antiderivative of the antiderivative (or the integral of the integral).
16. Be sure that you understand when the is positive, when it is negative and when it is zero (section 3.4)
If f(x) >0 and a<b then the integral is positive. What if f(x)>0 and a>b? What if f(x)<0 and a<b? What if f(x)>0 and a>b?
17. A simple definite integral problem for you. (section 3.4)
18. Given the graph of some function . If where “a” is some number (like 0 or 1 for example). Be able to look at the graph of and find things like for example (remember you are given the graph of NOT the graph of ) .
g(2) would be the net signed area under the graph of f(t) between “a” and 2. would be the “y” value of the graph of f(t) at 2 (recall the second part of the FTC). Think about this!
19. Another definite integral problem for you BUT this one is not as simple as #17. This time the antiderivative will be an inverse trig. function (so be sure that you can evaluate things like section 4.3 HW # 5 and 6).
20. Be able to solve simple related rates problems (section 2.5)
21 – 22. Be able to evaluate definite integrals using substitution (section 4.1). Note: Be sure to practice a VARIETY of problems involving different integrands (some involving trig functions (like HW #7 section 4.1 and some involving “e” like )
23. Given a function know how to tell things like….where it is continuous, where it has a max or min, where it is increasing or decreasing, where the derivative exists, where the second derivative is positive and negative. (sections 2.9, 2.10, 2.12)
24. If you are given the derivative of a function, be able to tell where the original function has relative max(s) and min(s)
(section 2.9). See Test #4 question #4 for an example.
25. Given the acceleration function of some particle along with two initial conditions (one for velocity and one for position), be able to find the position function. (section 3.1)
Remember that in order to find the velocity function just integrate the acceleration function (use the initial condition given for velocity to HELP you find the constant of integration that you get in your velocity function). THEN integrate your velocity function to obtain your position function AND now use the initial condition given for position to find the constant of integration for the position function.
26. Be able to find out where a given function is decreasing (section 2.9).
27. Make sure that you understand L’Hopital’s Rule and WHEN it applies! (section 2.11)
If you obtainwhen you do direct substitution then it applies so long as the function meets the criteria spelled out in L’Hopital’s rule (go read it)
28. Be sure that you understand the various properties of the definite integral that we covered (section 3.4)
29. Be able to evaluate definite integrals for basic sine and cosine functions (section 3.4)
30. Be able to evaluate a definite integral for a piece-wise defined function (section 3.4)
Remember that IF the piece-wise defined function is defined differently on one part of the interval that you are integrating over than an another then you must split the integral up using properties of the definite integral!
31. Be able to solve a related rates problem (section 2.5). Be sure to practice a variety of different types, including ones where angles change (obviously the one I select for your test must be easily doable without a calculator). In addition to your notes and HW you can try the following…
(Note: “Usually” when an angle is changing your “equation” will involve a trig function)
The top of a 25 foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute. When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?
Note: x was found to be 24 WHEN y=7 by using the Pythagorean theorem.
32. Be able to find points of inflection for a given function (section 2.10). Remember just because an x-value is a “possible” point of inflection, it doesn’t mean that it actually is.
Example: What is the Point of Inflection for ?
33. Be able to find limits at infinity! (section 2.11) Here are a couple of problems for you to practice BUT be sure to review your notes and HW!!!
Find
Find
Be sure to review your notes on limits at infinity. There are QUICK and easy ways to do these type of problems!
34. Be able to apply the extreme value theorem to find an absolute max of absolute min value of a function on a given interval. (section 2.7)
Example: What is the maximum value of the function ?
35. Be able to determine intervals on which a function is increasing or decreasing (section 2.9).
Example: On what interval(s) is decreasing?
Take the derivative and find the critical numbers . Break up the domain of f(x), your table will have 3 intervals (since you have two critical numbers). The sign of the derivative is only negative on the interval from -3 to 2 so that is the only interval on which the original function is decreasing.
36 and 37. Review over the rectilinear motion problems from section 2.9
Example: A particle moving along a horizontal line has position function . Find it’s acceleration function. When does the object speed up and slow down? When does the object change direction?
Note: The one on your final exam will work out MUCH nicer (and quicker) than this one BUT this still gives you one to practice that you haven’t seen (you can find others in your notes and HW etc.). Critical numbers will be 0, ½, and 1 (places where the velocity is zero….i.e. the only places where the object MIGHT change direction). Your PPOI will be
(your acceleration function is a quadratic that doesn’t factor so use the quadratic formula…obviously you won’t have THIS issue on the Final Exam as no calculators are allowed on this part).
38. Be able to find the value of “c” guaranteed by the Mean Value Theorem for derivatives (section 2.8)
39. If c is a positive real number, find an expression for in terms of c. What if c were a negative real number?
Draw a picture of . If c is a positive number then but if c is a negative number then . Use your knowledge of this type of “net signed area accumulator” function to derive expressions for the “net signed” are under your graph.
Note: If c is negative you should come up with whereas when c is positive you should come up with HOWEVER since c is positive isn’t = anyway?
40. Be able to evaluate a definite integral involving an absolute value function. (like section 3.4 HW # 8)
41 – 45. Be sure that you understand Newton’s Method (section 2.6) , the first derivative test (section 2.9) , the second derivative test (section 2.100, the mean value theorem for derivatives (section 2.8) and the mean value theorem for integrals (section 3.4). In particular make sure that you know exactly what each of these is used for in Calculus!
46 – 50. Several indefinite integral problems. Most of these are VERY quick and require little if any work at all! Be sure to know ALL of the integration formulas and rules that we have covered in this course!
THIS TEST WILL FAVOR THE EXTRAORDINARILY PREPARED STUDENT!