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

 

 

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