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

The Unit Circle

Here is a useful image of the unit circle labeled with the “special angles” and the coordinates of the corresponding points on the unit circle:

unit-circle-trig

(via http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215.htm)

This image is useful since you can use it to find the sine and cosine of any of the given angles, using the definitions of sin t and cos t as the y- and x-coordinates, respectively, of the point on the unit circle corresponding to the angle t:

186px-Unit_circle.svg

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

Quiz #5: Take-home, due Tues March 26

I handed out a take-home quiz at the end of class on Friday, which is due at the beginning of class this Tuesday (March 26).

I’ve uploaded the pdf of the quiz to the Files, for those of you that missed class Friday (or in case you need to print out another copy).

Study the example we did at the end of class Friday if you get stuck on the quiz. You can also look at Example 10.7(a) in the textbook (pp136-137).

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)

WebWork Hints: Shifting a Circle & Using Logarithms

There are two WebWork exercises that go beyond what we’ve covered in class this semester (although you should’ve seen this material in MAT1275 or an equivalent algebra course).  Hence I will count these two exercises as extra credit, but thought I’d also provide some hints to help you solve them:

Functions – Translations: Problem 4: In this exercises, you are given at equation of the form
$latex x^2 + y^2 = r^2$

The graph of this equation is a circle centered at the origin (0,0) with radius r.

In general, the equation of a circle centered at a point (h,k) with radius r is (x-h)^2 + (y-k)^2 = r^2 .

Use that to figure out the solution to the WebWork exercise, given that you are asked to shift the circle, and hence shift the center of the graph from (0,0) to another point (h,k).

Functions – Inverse Functions: Problem 10: In this exercise you are given a function f(x) involving e^(2x) terms, and you are asked to find the inverse function f^{-1}(x).

In general, as we outlined in class and as you should’ve done on the previous exercises in this HW set, in order to solve for the inverse function, set up the equation y = f(x), and then solve for x in terms of y.

In order to do that in this exercise, where the x terms are in the exponent, you need to use the natural log function.

Here’s a simpler example:

Say f(x) = 7e^{2x}.  Then in order to find f^{-1}(x):

y = 7e^{2x}

y/7 = e^{2x}

and now take the natural log of both sides:

ln (y/7) = ln (e^{2x})

On the LHS, ln (e^{2x}) = 2x (this is by the definition of ln, which is the inverse function of e^x!)

Hence,

ln (y/7) = 2x

x = ln (y/7)/2

and so in this example,

f^{-1}(x) = ln (x/7)/2

 

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

First OpenLab Assignment – Introduce Yourself

Your first OpenLab assignment is to introduce yourself to your classmates.  This assignment is due Friday, February 1, at the start of class.  Completing this assignment will earn you a point towards the participation component of your course grade. Late submissions will receive partial credit.

Assignment. Write a comment in reply to this post (scroll to the bottom to find the “Leave a Reply” box–if you’re viewing this from the site’s homepage, you will need to click on the post’s title above, or click on the Comments link to the left):

In a brief paragraph (3-5 sentences), introduce yourself in whatever way you wish (what do you want your classmates to know about you?  Some ideas: where you’re from, where you live now, your major, your interests outside of school, etc.)