Guide to the Final Exam Review

As I have been saying over the course of the semester, you should try to work through the Final Exam Review exercises before the final exam (coming up on Tuesday May 21).

Below is a guide to the exercises/topics on the FER, with similar exercises from the quizzes and exams listed. Look at the solutions of the exams and quizzes for further review as you work through the FER exercises! (All the solutions have been uploaded to the Files.)

Specifically, here’s what I recommend you do to prepare for the final:

  • As you go thru the list of FER questions below:
    • Try to do the FER exercise
    • If you can’t remember how to do that kind of problem, go through the solutions of the corresponding Quiz and Exam questions
    • Then try the Final Exam Review exercise again, using the quiz and/or exam solution as a guide
    • You can also look at these YouTube videos, which work thru some of the FER exercises

#1: Polynomial and rational function inequalities

  • Use the graph of the function on the LHS of the inequality to find the interval(s) where the function is greater than or less than 0 (i.e., where the graph above or below the x-axis).
  • For inequalities involving a quadratic polynomial, such (a)-(c), solve for the roots of the quadratic by factoring (or by using the quadratic formula). This gives you the x-intercepts of the graph, i.e., it tells you exactly where the graph crosses the x-axis.
  • For inequalities involving a rational function, such as (d), solve for the x-intercept(s) (by solving where the numerator equals 0), but you’ll also need to solve for the vertical asymptote(s) (by solving where the denominator equals 0). Then look at the graph of the rational function.
  • See Exam 2, #1 & #4; and Quiz #6

#2: Absolute value inequalities

  • If the inequality is \lvert mx + b \rvert < c, then solve - c < mx + b < c
  • If the inequality is \lvert mx + b \rvert > c, then solve mx + b < -c and solve mx + b > c
  • See Quiz 1, #2; Exam 1, #2; Exam 3, #1

 

#3: Rational Functions

  • Find the x- and y-intercepts algebraically, the domain, the vertical and horizontal asymptotes
  • See this for a summary for how to do this algebraically
  • Graph the function using a graphing calculator and the information above
  • See Exam 2, #4 & Quiz #6

 

#4: Difference Quotients

  • Find f(x+h) in order to set up and simplify the difference quotient \frac{f(x+h) - f(x)}{h} (in particular for quadratic polynomials, i.e., f(x) = ax^2 + bx + c)
  • See Exam 1, #3 ; Exam 3, #2

 

#5: Polynomials

  • Find the roots of a polynomial algebraically (using a combination of techniques: by identifying an integer root at x=c and using long division by (x-c); by factoring; by using the quadratic equation)
  • Graph the function using a graphing calculator and the roots
  • See Quiz 5 and Exam 2, #3

 

#6: Vectors

  • Find the magnitude and direction angle of a vector
  • See Example 22.4 on pp301-302 of the textbook
  • work thru “Vectors – Magnitude and Direction” WebWork

#7: Skip this since we didn’t have time to cover this topic

#8: Properties of Logarithms

  • Simplify logarithmic expressions using the properties of logarithms
  • See Exam 3, #4

 

#9: Graphing logarithmic functions

  • Finding the domain, asymptote and x-intercept of a logarithmic function, and sketching its graph
  • See Exam 3, #7

 

#10: Amplitude, period, phase shift of a trigonometric function

  • See Exam 4, #5 & #6

 

#11: Solving trigonmetric equations

  • See Exam 4, #1 & #2

 

#12 and #13: Exponential population growth

  • Write down the function P(t) = P_0 (1+r)^t given the initial population P_0 and the rate r at which the population is growing (or decreasing, as in #12!)
  • Solve for the time it takes for the population to grow (or decrease) by a certain multiple (i.e., to double to 2*P_0); this involves solving an exponential equation by using logarithms
  • See Exam 4, #3 & #4

 

#14: Find the inverse of a given function

  • Given y = f(x), switch x and y, and solve for y in terms of x
  • See Exam 1, #5

#15: Sums of arithmetic sequences

  • Given an arithmetic sequence, where each successive term is found by adding a constant “difference” d the previous term (i.e., a_{i+1} = a_i + d for all i > 1), identify the initial term a_1 and the constant d
    • For example: given the sequence 25,21,17,13,9,5, \ldots, we see that a_1 = 25 and d = -4
  • Use the following formulas to calculate the sum of the 1st k terms of the sequence:
    • the k-th term of the sequence a_k is given by the formula a_k = a_1 + (k-1)*d
    • then use the formula for the sum of the 1st k terms of the sequence: \Sigma_{i=1}^k a_i = \frac{k}{2}(a_1 + a_k)
  • For example, to calculate the sum of first 83 terms of the arithmetic sequence 25,21,17,13,9,5, \ldots:
    • first we calculate that the 83rd term in the sequence is a_{83} = a_1 + (k-1)*d = 25 + (83-1)*(-4) = 25-328 = -303 and so
    • the sum of the first k = 83 terms is \Sigma_{i=1}^{83} a_i = \frac{83}{2}(25 - 303) = -11,537
  • See Exam 4, #7
  • See also Example 23.15(c) on pp321-322 of the textbook
  • The formulas above are on p318 and p321, respectively

 

#16: Sums of infinite geometric sequences

  • Given a geometric sequence with initial term a_1 and constant “ratio” r, i.e., each successive term is found by multiplying the previous term by r (so a_{i+1} = a_i * r for all i > 1), identify the constant r
  • Then use the following formula to calculate the sum of an infinite geometric sequence: \Sigma_{i=1}^k a_i = a_1 * \frac{1}{1-r}
  • For example, given the sequence -6, 2, -\frac{2}{3}, \frac{2}{9}, -\frac{2}{27}, \ldots, we see that r =-\frac{1}{3}, with a_1 = -6
  • Hence, 1-r = 1 - (-1/3) = 1 + (1/3) = 4/3, and so \frac{1}{1-r} = 3/4, and thus the sum of this geometric series is a_1 * \frac{1}{1-r} = -6*(3/4) = -18/4 = -9/2.
  • See Exam 4, #8
  • See also Example 24.10(c) on pp332-333 of the textbook

#17: Skip this since we didn’t have time to cover this topic

 

 

Take-home Exam #4

Here are the notes I wrote on the board yesterday at the end of class, as a guide to the take-home exam:

  • Questions #1 & #2 (solving trigonometric equations using the unit circle)
    • study the examples we did in class
    • work thru the “Trigonometry – Equations” WebWork
    • work thru Final Exam Review #11
  • Questions #3 & #4 (exponential growth/decay models)
    • review the “Exponential Functions – Growth and Decay” WebWork
    • look at textbook (Ch 15)
    • work thru Final Exam Review #12 & #13
  • Questions #5 & #6 (graphing trig functions)
    • review the “Trigonometry – Graphing Amplitude” & “Trigonometry – Graphing Period “WebWork sets
    • look at textbook (Ch 17)
    • work thru Final Exam Review #10
  • Questions #7 & #8 (sums involving arithmetic and geometric sequences)
    • study the examples we did in class
    • work thru the “Sequences – Arithmetic” and “Series – Finite Arithmetic” WebWork sets
    • work thru Final Exam Review #15 & #16

Midterm Exam #3 – Tues April 30

Hope you’re all having a good spring break. A reminder that we will have our third midterm exam on Tuesday, April 30 (i.e., during our first class after spring break). This exam will cover the material on exponential and logarithmic functions we have discussed over the past few weeks, but will also cover some material from earlier in the semester (in particular, there may be questions on absolute value inequalities, difference quotients, and inverse function; see below for details).

To prepare for the exam:

  • review the following WebWork assignments/exercises:

    • “Exponential Functions – Graphs”: #1-6
    • “Logarithmic Functions – Graphs”: #1-2
    • “Logarithmic Functions – Equations”: #1-4
    • “Logarithmic Functions – Properties”: #1-4
    • “Exponential Functions – Equations” (this set will be due Friday 5/3: even though these are due after the exam, do at least the following initial exercises before the exam): #1-3
  • review solutions for Quiz #7 (solutions have been uploaded to Files)

 

  • review Exam #1 (solutions have been uploaded to Files), specifically:
    • #2 (absolute value inequalities)
    • #3 (difference quotient)
    • #4 (finding an inverse function)
  • write out solutions for the following Final Exam Review (FER) exercises (download the FER pdf here)
    • #2 (solving absolute value inequalities)
    • #4 (finding the difference quotient of a given function)
    • #8 (simplifying/expanding logarithmic expressions by applying the properties of logarithms)
    • #9 (finding the domain, asymptote(s), and x-intercepts of a simple logarithmic function, and sketching its graph)
    • #12 & #13: solving word problems involving exponential growth or decay
    • #14: finding the inverse of a function
  • most of the exercises on Exam #3 will be similar to the FER exercises listed above
  • I will count any FER exercise solutions you hand in as extra-credit towards your homework score; you can hand these in at any time before the end of the semester

Midterm Exam #2 – Tues, April 2

We will have our second midterm exam on Tuesday, April 2. To prepare for the exam:

  • review the following WebWork assignments/exercises:

    • “Polynomials – Graphs”: #1-3
    • “Polynomials – Theory”: #1, 2
    • “Polynomials – Inequalities”: all (#1-4)
    • “Rational Functions – Intercepts”: #1-6
    • “Rational Functions – Domains”: #1-5
    • “Rational Functions – Inequalities”: #1-3
    • “Polynomials – Division” (this set is due Friday 4/5: even though these are due after the exam, do at least the following initial exercises before the exam): #1, 2
    • “Rational Functions – Asymptotes” (this is also due Friday 4/5 but do at least the following initial exercises before the exam): #1-3
  • you can study the WebWork solutions (for any given WebWork set, click on “Generate PDF or TeX Hardcopy for Current Set” and then select the “Correct Answers” & “Solutions” options)
  • in addition to reviewing the solutions, you can click on “Show Me Another” to generate a new exercise that you can practice with (solutions are provided for “Show Me Another” exercises)
  • review solutions for Quiz #4, Quiz #5 & Quiz #6 (solutions will be uploaded to Files)
  • review the following Final Exam Review exercises (download the “FER” pdf here)
    • #1 (polynomial & rational inequalities)
    • #3 (analyzing & graphing rational functions: domains, intercepts, vertical asymptotes)
    • #5 (analyzing & graphing polynomials: roots, intercepts, relative maxima/minima)

Midterm Exam #1 – Review

We will have our first midterm exam next Tuesday (Feb 26).  To prepare for the exam:

  • start by finishing the open WebWork assignments (“Functions – Translations” & “Functions – Inverse Functions”) which are due Monday (Feb 25) at 2p
    • a few of the more advanced exercises go beyond what will you will be asked on the exam, and I will count those exercises count as extra credit; see here for a list of which exercises are extra credit, and see below for which exercises to make sure you understand for the exam)
  • the WebWork sets are due before the exam so that you can study the WebWork solutions (for any given WebWork set, click on “Generate PDF or TeX Hardcopy for Current Set” and then select the “Correct Answers” & “Solutions” options)
  • review the following WebWork exercises; in addition to reviewing the solutions, you can click on “Show Me Another” to generate a new exercise that you can practice with (solutions are provided for “Show Me Another” exercises)
    • Interval Notation: #3 & #4
    • Absolute Value Inequalities: #6 & #7
    • Functions – Notation: #1-5
    • Functions – Operations: #2-4
    • Functions – Difference Quotient: #1 & #2
    • Functions – Translations: #1-3
    • Functions – Inverse Functions: #1-8
  • review Quiz #1 & Quiz #2 (solutions have been uploaded to Files)
  • I will have extra office hours Monday morning (Feb 25), 10a-12p for exam review