Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
The Math 1050 committee determined which learning objectives from each section were important
enough to appear on the final exam. We tried to choose the 2 or 3 most important topics in each section. We
then wrote questions to address each objective. Most objectives have 3 questions: easy, medium, and hard. The
Math 1050 committee will develop a common final exam for Math 1050 Spring 2017 taken exclusively from
this set of question types.
The format of the final exam will be 20 questions. These questions will be a mixture of easy, medium,
and hard questions, balanced to create an exam well prepared students can finish within 1hr 50min. 12 of the
questions will be required to appear on all common final exams. These will be labeled. The remaining 8 of these
questions may be used, or may be traded for material the instructor emphasized more that semester and would
like to pose questions on. Instructors are free to substitute required questions with harder questions, with
discretion and consideration that they are not making their entire final exam too hard for their students.
The format of this review for each section is: first the important learning objectives that will appear on
the final exam are listed, then examples of all possible problem types we will consider when writing the
common final exam are listed. In some sections, we identified easy, medium, and hard levels. In other sections
we did not. Answers are not provided. We thought teachers could solve all of these well enough. Answers to
specific questions are available upon request.
Well prepared students should be able to address all of these topics with accuracy and fluency.
This document is being provided in the hopes that it will calm any fears regarding the common final,
and provide enough information that instructors can tailor their instruction to meet these expectations. Please,
address any questions or concerns to the Math 1050 committee.
With Best Wishes,
Jennifer Hooper – Chair – [email protected]
Ya Li – [email protected]
Yingxian Zhu – [email protected]
David Brandt – [email protected]
Colin Brinkerhoff - [email protected]
Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
Functions
A student is able to:



Compute the difference quotient for a variety of types of functions.
Use transformations to sketch the graphs of linear, power, root, reciprocal and absolute value functions.
Find the inverse of a function formally (algebraically) and graphically.
QUESTION TYPES
1. Find the difference quotient,
a. f(x) = – 2x + 5
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
, and simplify.
b. f(x) = x2 – 2
1
c. 𝑓(𝑥) = 𝑥+3
2. Use transformations (or translations) to graph the given function.
a. 𝑓(𝑥) = −√𝑥 + 3 − 1
1
b. 𝑓(𝑥) = 2 (𝑥 − 4)2 + 3
c. 𝑓(𝑥) = −3|𝑥 + 2| − 4
1
d. 𝑓(𝑥) = 𝑥+2 − 3
3. The following functions are one-to one. Write an equation of the inverse function.
a. 𝑓(𝑥) = −5𝑥 + 4
b. 𝑓(𝑥) = 4𝑥 3 + 2
5
c. 𝑓(𝑥) = −𝑥+2
𝑥−5
d. 𝑓(𝑥) = 𝑥+1
Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
4. The graph of a function is given. Determine if it has an inverse. If yes, sketch the inverse function.
a.
Source: https://www.softmath.com/tutorials-3/relations/articles_imgs/4103/invers61.gif
b.
Source: https://vt-s3-files.s3.amazonaws.com/uploads/problem_question_image/image/31525/Q4_2.png
c.
Source: https://vt-s3-files.s3.amazonaws.com/uploads/problem_question_image/image/31515/Q3.png
Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
Polynomial and Rational Functions
A student is able to:
 Graph a polynomial, showing x- and y-intercepts and proper end behavior.
 Use the Rational Zeros Theorem, Descartes’ Rule of Signs, and the Upper and Lower Bounds Theorem
in finding zeros of polynomials.
 Solve polynomial equations.
 Graph a rational function, showing intercepts and asymptotes.
QUESTION TYPES
5. Easy: p(x) = x3 – 7x + 6
a. Use the Rational Zero Theorem to determine all possible rational roots
b. Use Descartes’s rule of signs to determine the number of positive and negative roots
c. Factor the polynomial
d. Sketch the graph, showing intercepts and proper end behavior
6. Medium: p(x) = 6x3 + 17x2 + x – 10
b. Use the Rational Zero Theorem to determine all possible rational roots
c. Use Descartes’s rule of signs to determine the number of positive and negative roots
d. Factor the polynomial
e. Sketch the graph, showing intercepts and proper end behavior
7. Hard: p(x) = 2x6 – x5 – 13x4 + 13x3 + 19x2 – 32x + 12
a. Use the Rational Zero Theorem to determine all possible rational roots
b. Use Descartes’s rule of signs to determine the number of positive and negative roots
c. Factor the polynomial
d. Sketch the graph, showing intercepts and proper end behavior
8. Verify that the following values are upper or lower bounds for the given polynomials.
a. p(x) = x4 – 3x2 – 2x + 5, 3 upper bound, -2 lower bound
b. p(x) = 2x5 + 5x4 + x3 – 3x + 4, 1 upper bound, -3 lower bound
9. Solve the polynomial equations
a. 6x3 + x2 – 5x – 2 = 0
b. x5 – 2x4 + 3x3 + 2x - 3 = x4 + 4x3 – 7x2 + 2x + 1
𝑥+2
10. Medium: Graph the rational function 𝑓(𝑥) = 𝑥 2 −4𝑥+3.
a.
b.
c.
d.
e.
Show and label the vertical asymptote(s), if any.
Show and label the horizontal asymptote(s), if any.
Show and label the x and y intercepts, if any.
Write the domain in interval notation.
Graph the function.
Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
𝑥 2 +𝑥−2
11. Hard: Graph the rational function 𝑓(𝑥) = 𝑥 3 −𝑥 2 −6𝑥. Show and label all asymptotes, intercepts, and
hole(s), if any.
12. Medium: Find all the asymptotes (vertical, horizontal, and/or slant) of the following rational function.
𝑓(𝑥) =
𝑥 3 −2𝑥 2 −3𝑥
𝑥 2 −4
.
Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
Exponential and Logarithmic Functions
A student is able to:
 Graph exponential and logarithmic equations.
 Solve logarithmic and exponential equations.
QUESTION TYPES
13. Starting with the basic function f (x) = log 2 x , use transformations, as needed, to answer the following
questions for the function g ( x)  1  log 2 ( x  4) .
a. Identify the y-intercept of g(x) , if any:
b. State the equation of the asymptote of g(x) ,
if any:
c. Find the domain and the range of g(x)
using interval notation.
d. Sketch the graph of g(x) .
14. Starting with the basic function: f (x) = 2 x , use transformations, as needed, to answer the following
questions for the function g(x) = 2 x+2 -1.
a. Find the y-intercept of g(x) , if any:
b. Find the asymptote of g(x) , if any:
c. Sketch the graph of g(x) .
15. Solve the equation log 6 (𝑥 + 2) + log 6 (𝑥 − 3) = 1. Please, give exact answers.
16. Solve the logarithmic equation log 3 9 - log 3 (x + 2) = 2 . Please, give exact answers.
17. Solve the exponential equation e2 x + 5ex - 6 = 0 . Please, give exact answers.
Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
Systems of Equations and Inequalities
A student is able to:
 Solve a system of nonlinear equations in two variables.
 Write the solutions of a linear system of equations with infinitely many solutions.
 Graph the solution set of a system of inequalities.
 Compute the partial fraction decomposition of a rational function when the denominator is a product of
linear factors or distinct quadratic factors.
QUESTION TYPES
18. Solve the following system of nonlinear equations in two variables.
a. {
𝑥 2 + 𝑦 2 = 10
3𝑥 + 𝑦 = 0
b. {
𝑥 2 + 𝑦 2 = 16
𝑥2 − 𝑦2 = 2
c. {
𝑥2 + 𝑦2 = 9
𝑥2 − 𝑦 = 3
19. Solve the system. If the system has one unique solution, write the solution set. If the system of equations
has infinitely many solutions, write the general form of the solution. If the system of equations has no
solutions, so state.
a. {
2𝑥 − 𝑦 = −10
−4𝑥 + 2𝑦 = 20
−3𝑥 + 4𝑦 − 𝑧 = −4
𝑥 + 2𝑦 + 𝑧 = 4
b. {
−12𝑥 + 16𝑦 − 4𝑧 = −16
c. {
𝑥 + 2𝑦 + 4𝑧 = 3
𝑦 + 3𝑧 = 5
𝑥
− 2𝑧 = −7
20. Graph the solution set. Label all vertices. If there is no solution, indicate that the solution set is the
empty set.
1
a. {
𝑦 ≥ 3𝑥 − 2
𝑦 <𝑥−4
𝑦 ≤ −𝑥 2 + 7
b. { 𝑦 ≤ −𝑥 + 5
𝑦>1
𝑥 2 + 𝑦 2 ≤ 25
c. {
𝑦>2
𝑥<3
Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
21. Set up the form for the partial fraction decomposition. Do not solve for A, B, C, and so on.
2𝑥 5 + 3𝑥 3 + 4𝑥 2 + 5
𝑥(𝑥 + 2)3 (𝑥 2 + 2𝑥 + 7)2
22. Find the partial fraction decomposition of the following rational functions.
a.
b.
c.
d.
−𝑥−37
(𝑥+4)(2𝑥−3)
6𝑤−7
𝑤 2 +𝑤−6
17𝑥 2 −7𝑥+18
7𝑥 3 +42𝑥
4𝑥 3 −4𝑥 2 +11𝑥−7
𝑥 4 +5𝑥 2 +6
Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
Matrices and Determinants
A student is able to:
 Use Gauss-Jordan elimination to solve a system of equations.
 Solve a system of equations using an inverse matrix.
 Use Cramer's rule to solve a 2x2 or 3x3 system of equations.
QUESTION TYPES
23. Solve the following using Gaussian elimination or Gauss-Jordan elimination.
a. {
2𝑥 − 7𝑦 = −41
3𝑥 − 9𝑦 = −51
𝑥
+𝑧 =5
𝑦 + 2𝑧 = 8
b. {
𝑥 + 4𝑦
=1
c. {
−2𝑥 + 5𝑦 − 4𝑧 = −4
𝑥 − 2𝑦 + 𝑧 = 3
𝑥 − 5𝑦 + 9𝑧 = −5
24. Solving the following system of equations using the inverse of the coefficient matrix.
3
a. Given 𝐴 = [
4
b. {
3𝑥 + 7𝑦 = −5
7 −1
−9 7
], 𝐴 = [
], solve {
4𝑥 + 9𝑦 = −7
9
4 −3
−4𝑥 − 3𝑦 = −4
6𝑥 + 5𝑦 = 8
𝑥 − 2𝑦 + 2𝑧 = −12
c. {2𝑥 − 3𝑦 + 𝑧 = −10
𝑦−𝑧=6
25. Solve the system using Cramer’s rule.
a. {
2𝑥 + 10𝑦 = 11
3𝑥 − 5𝑦 = 6
𝑥
−𝑧 = 7
3𝑦 + 𝑧 = 4
b. {
−𝑥 + 𝑦
=1
2𝑥 − 5𝑦 + 𝑧 = 11
c. {3𝑥 + 7𝑦 − 4𝑧 = 8
𝑥 − 9𝑦 + 2𝑧 = 4
Math 1050 Common Final Exam Question types – Effective Spring 2017 (Revised 4/25/17)
Sequences and Series
A student is able to:
 Find a formula for the nth term of an arithmetic sequence given several terms.
 Find the nth partial sum of an arithmetic or geometric sequence.
 Calculate the sum of an infinite geometric series.
 Use Pascal’s Triangle to expand a binomial. Use the Binomial Theorem to expand a binomial.
 Find a particular term of a binomial expansion.
QUESTION TYPES
26. Given the arithmetic sequence
3, 7, 11, 15, 19, …
a. Find an expression for the nth term 𝑎𝑛
b. Find the 100th term 𝑎100
c. Find the sum of the first 100 terms.
27. A theater has 20 rows of seats. The first row has 15 seats, the second row has 17 seats, and each of the
remaining rows has two more seats than the previous row. How many seats are in the theater?
28. Given the geometric sequence
1 1 1
1, − , , − , …
2 4 8
a. Find an expression for the nth term 𝑎𝑛
b. Find the 6th term 𝑎6
c. Find the sum of the first 10 terms of the geometric sequence. Just set it up. Do not simplifyneed calculator.
1
1
29. Find the sum of the infinite series 1 + 5 + 25 +
1
125
+ ⋯. If the sum does not exist or is infinite, so state.
30. Find the sum of the infinite geometric series:
S =
900 – 720 + 576 – 460.8 + ∙ ∙ ∙
If the sum does not exist or is infinite, so state.
31. Use either the binomial theorem or Pascal’s Triangle to expand the binomial. Be sure to simplify each
term of the expansion.
(𝑥 − 2)5
(2 x + 1)5
32. Find the coefficient of the term containing 𝑏 4 for the binomial:
(𝑎 + 3𝑏)6
(2𝑎 − 𝑏)10