Item bank for Math 120 – Spring 2008.
These sample problems were prepared by a committee of NB high school teachers,
as part of a workshop held at the University of New Brunswick, March 13 -- 14, 2008.
Sequences and series
One point problems.
A sequence is defined by and . Find the values of and .
A sequence is defined by and . Find the values of and .
Find the sum of the following geometric series: 100 + 25 + 6.25 + … .
Find the common difference for the arithmetic sequence having and .
Fill in the missing terms in the following arithmetic sequence: 3, __, __, __, 81.
Fill in the missing terms in the following geometric sequence: 3, __, __, __, 81.
Look for a pattern in this sequence and write a recursive definition of the sequence.
2, 6, 18, 54 … .
Find for the sequence defined recursively by: , .
Express using sigma notation.
Write as a fraction.
Evaluate .
Evaluate .
Evaluate .
Determine the value of .
Determine the value of .
Three point problems.
Given the arithmetic series 1 + 6 + 11 + : : : + 76.
(a) Find the sum.
(b) Write a formula for the series using sigma notation.
An arithmetic sequence has and . Find
(a) the common difference.
(b) the first term.
Given the geometric series 2+4 + 8 + … + 1024.
(a) Find the sum of the series.
(b) Express he series using sigma notation.
Your Mom offers you two options for allowance on a trial basis of 20 weeks.
Option A: $30/week for 20 weeks.
Option B: Start with 1c the first week, 2c the second week, 4c the third week and so on, doubling each week.
Calculate the total allowance earned for the entire 20 weeks for each option.
(Show work. The numerical values alone are not sufficient.)
Use six rectangles to approximate the area under the curve from to .
Ahmed was assigned the following problem.
Use mathematical induction to prove that .
Fill in the gap in Ahmed’s proof:
Let be the statement that .
Basis step: states that .
So is true.
Induction step: Assume that is true for some .
That is, assume that .
Then
Therefore implies .
Conclusion: is true for all .
Five point problems
As it aged, a maple tree produced sap according to the pattern shown in the table below.
Year / 2001 / 2002 / 2003 / 2004Sap (litres) / 60.000 / 57.000 / 54.150 / 51.4425
t1 / t2 / t3 / t4
(a) Do the data follow an arithmetic or a geometric pattern?
(b) Write down a formula for .
(c) Assuming that the pattern continues, in what year will production be approximately 17.5 litres?
(d) If the tree were to live a longtime, what would be its total sap production from 2001 onwards?
A computer software company formed a committee of five people to spread the word about a new
feature. Each person on the committee e-mailed 3 individuals (cycle 1) who were each asked to
contact 3 more individuals (cycle 2). This pattern continued, and new individuals were contacted at
each cycle (i.e. nobody received two e-mails).
Let represent the number of people contacted at the nth cycle.
(a) What is the value of?
(b) Write down a formula for.
(c) On which cycle were 32 805 people contacted?
(d) Find the total number of people contacted after the tenth cycle.
Functions I
One point problems.
If and then solve for x: .
. Find .
Joe investigated a polynomial function and found that .
For what values of x is the original function f increasing?
and . Find the value of .
Given and , find a formula for .
The functions f(x) and g(x) are graphed below. Find g(f(1)).
Write down an equation for the cubic polynomial graphed below.
Write down an equation for the cubic polynomial graphed below.
Frosty the Snowman's spherical head is melting. Determine a formula for the rate of change of the volume of his head with respect to its radius.
(Volume of a sphere: , where r represents radius.)
The graph of the function appears on the left.
Sketch the graph of on the other grid.
Three point problems
Write in factored form: .
Write in factored form: .
Solve: .
Solve: .
Solve: .
Sketch a possible graph of a 5th degree polynomial function with the given property, if possible.
If not possible, explain why.
(a) Exactly one real root.
(b) No real roots.
(c) Exactly 4 distinct real roots.
Consider the function .
Use the definition of derivative to find a formula for .
Consider the function
(a) Sketch the graph of this function.
(b) Is the function continuous at ?
Consider the function
(a) Sketch the graph of this function.
(b) Find .
(c) Find .
At what point is the graph of parallel to the line ?
Write in the form .
Solve for real x and y: .
The complex number z is written in polar form as .
(a) Write in polar form.
(b) Plot the point .
Five point problems
Consider the function .
(a) Use the definition of derivative to find a formula for .
(b) Find an equation for the tangent line to the graph of at the point (3, 22).
Consider the function .
(a) Use the definition of derivative to find a formula for .
(b) Find an equation for the tangent line to the graph of at the point (3, 36).
Given .
(a) Find all critical points.
(b) Find all intervals in which the function decreases.
(c) Sketch the graph of the function on the axes below, being careful to show all intercepts and critical points and the basic shape of the graph.
Given .
(a) Find all critical points.
(b) Find all intervals in which the function decreases.
(c) Sketch the graph of the function on the axes below, being careful to show all intercepts and critical points and the basic shape of the graph.
Functions II
One point problems.
Find the domain of the function .
, . Find .
If , find .
If then .
What is the domain of the function ?
Give an example of a rational function with domain .
Find the limit of as x approaches .
Find .
Evaluate .
Determine the equation of the oblique asymptote of the graph .
Determine the equation of the oblique asymptote of the graph .
Simplify .
Solve for x: .
The graph of the function appears on the left.
Sketch the graph of on the other grid.
Three point problems.
Solve: .
Solve: .
What are the domain and range of the function ?
Simplify, and state restrictions: .
(Algebraic solution required.) Solve for x: .
Consider the function .
Write down:
(a) Equations for all horizontal asymptotes:
(b) Equations for all vertical asymptotes:
(c) The values of all x-intercepts.
Five point problems
Given .
(a) Find all xand yintercepts.
(b) Find the domain and range of the function.
(c) Find all asymptotes.
(d) Sketch the graph of the function on the axes provided.
(Plot key points, and show the basic shape of the graph.)
Given
(a) Find all xand yintercepts.
(b) Find the domain and range of the function.
(c) Find all asymptotes.
(d) Sketch the graph of the function on the axes provided.
(Plot key points, and show the basic shape of the graph.)
(Algebraic solution required.) Solve for x:.
(Algebraic solution required.) Solve for x:.
Complex numbers
One point problems
What is the modulus of the complex number graphed below?
What is the modulus of the complex number graphed below?
If and , find .
Express in polar form: .
Simplify the expression .
Write in the form .
Three point problems.
Solve: .
Find all solutions of the equation .
Simplify .
(Show work.) Solve for real x and y: .
The complex number z is written in polar form as
(a) Write in polar form.
(b) Plot the point on the complex plane.
Five point problems
Evaluate .
Evaluate .
(a) Write the complex number in polar form.
(b) Find all solutions of the equation .