Math_105_003_Spring_2018_FINAL_EXAM_Study_Guide

Math 105-003 Spring 2018 - Preston

5 May 2018 Study Guide for the Comprehensive Final Exam Page 1 of 3

Study Guide for Comprehensive Final Exam

Date/Time: 10:30 AM – 1:15 PM Wed 9 May 2018

Robinson B104 (our classroom)

General Information About the Exam:

 Same general rules apply (see syllabus):

o In particular, no calculators of any kind may be used

 Without going into detail, we’ve had an issue with the identity of person(s) taking an exam.

EVEN IF I KNOW YOU, when you turn in your exam, please wait for me to mark it on my list and

BE PREPARED TO SHOW YOUR GMU ID

 Please don’t be insulted, this is not personal – I’m just doing my job. I’d rather not have to

be so formal but I’d also rather not have a student send someone else to take their exam

for them and think that I’m an idiot and can’t figure it out. So … this is the result – everyone

will need to be prepared to prove their identity.

 There are 8 numbered problems (three of them have 2 parts) totaling 115 points (15% built-in

extra credit)  this exam will be only slightly longer than one of our mid-terms. However, you

will have 2 hours and 45 mins compared to the 1 hour and 50 minutes for the mid-terms.

o Many of the problems are extremely similar to problems on our 3 mid-term exams and/or

the previous final exams, or previous mid-term exams.

 All but one problem has appeared either exactly or nearly identically on prior exams

(and often they came from homework exercises or even numbered book exercises)

 The

 SHOW ALL OF YOUR WORK!!!

o I’ve gotten numerous answers on mid-term exams with no “solutions”, i.e. no work to back

them up. Those have typically gotten much less than full credit, sometimes no credit. On

the final exam, I will be even more strict about this  NO WORK NO CREDIT, PERIOD

I can’t grade what I don’t see  no work no credit

Math 105-003 Spring 2018 - Preston

5 May 2018 Study Guide for the Comprehensive Final Exam Page 2 of 3

Some Things You Definitely Need to Know How to Do:

This is not necessarily in priority order or all inclusive

 Solve a quadratic equation and be able to determine when you will get one of the

three possible cases based on the discriminant 𝑫 = 𝒃𝟐− 𝟒𝒂𝒄

 Two real and distinct roots (𝐷 > 0)

 Two real and equal roots (𝐷 = 0)

 Two complex conjugate roots (𝐷 < 0)

 Build a polynomial from its roots and other given information (such as a point it

passes through to determine the “A” in the expression below)

 Standard factored form is: 𝒇(𝒙)= 𝑨(𝒙 − 𝒓𝟏)(𝒙−𝒓𝟐)(𝒙−𝒓𝟑)…(𝒙 − 𝒓𝒏) where A is an real

constant and 𝑟

1, 𝑟2, 𝑟3, … , 𝑟

𝑛are the roots, real or complex

 Standard form when multiplied out: 𝑓(𝑥)= 𝑎𝑛𝑥𝑛+ 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎2𝑥2+ 𝑎1𝑥 + 𝑎0

 Find all of the real and complex roots of a polynomial

 Know the key theorems of chapter 3 and how to use them, for example:

 Chapter 3.2

 Remainder Theorem

 Rational Zeros Theorem

 Intermediate Value Theorem

 Chapter 3.3

 Fundamental Theorem of Algebra, and especially the one right below it called

“Theorem”, and in particular the application to polynomials with real coefficients (all we

have studied really) captured in “Theorem” right before Example #3

 Conjugate Pairs Theorem and it’s corollary

 Rational Functions 𝑹(𝒙)=𝒑(𝒙)

𝒒(𝒙)

 How to find the domain, vertical asymptotes, horizontal OR oblique asymptotes, “holes” in the

graph, end behavior, x- and y-intercepts

 HOW TO SKETCH A CORRECT AND WELL-LABELED GRAPH OF 𝑅(𝑥) based on the

aforementioned data and analysis of the sign of the function within key intervals

 Finding the intersection of two functions (i.e. solving 𝒇(𝒙) = 𝒈(𝒙)) and/or solving

an inequality involving two functions (e.g. 𝒇(𝒙) ≥ 𝒈(𝒙))

 Be able to handle an unknown constant in an expression and treat it just like it

were a number, and given the parameters of the problem, develop an equation to

solve for the unknown constant

 Solve a logarithmic and/or exponential equation

Math 105-003 Spring 2018 - Preston

5 May 2018 Study Guide for the Comprehensive Final Exam Page 3 of 3

 Know how to use the Laws of Logarithms and Exponents to correctly simplify (or modify) the

equation in order to solve it

 Know the fundamental inverse relationship between the log and exponential functions of the

same base. That is: log𝑎𝑦 = 𝑥   𝑎𝑥= 𝑦 and CORRECTLY apply it

 Correct: log𝑎𝑎𝑥= 𝑥 eln 2= 2

 Incorrect: e3ln 2≠ 3 ∙ 2 = 6 NO!!!!!!!!! e3ln 2= eln 23= eln 8= 8

 Be aware that a log or exponential equation might be a “quadratic in disguise” and require

some algebraic manipulation to where you can substitute something like 𝑢 = 𝑎𝑥 or 𝑢 = log𝑎𝑥

to fully reveal its quadratic structure.

 Find the general solution to a trigonometric equation

 Just like the log and exponential functions above, a trig equation may also be in quadratic form

once an appropriate u-substitution is made

 Word Problems (e.g. Exponential Growth or Decay)

 Applying the Law of Sines and/or Cosines

 Given that you will not have a calculator, you may end up with a final answer that includes

expressions such as “sin 50 °" or "cos−1(…)"

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