The three equally-weighted benchmark tests will contribute
15% to your final course grade; each benchmark test will be weighted 5%.
Benchmark testing is the department's way of assuring that students
have achieved minimum levels of calculational competency. It is generally
expected that students who have successfully taken a semester of calculus
can do certain calculations by hand. Although we
are using computers and calculators throughout this course, you are
still expected to learn to do these hand calculations.
The first benchmark test has been scheduled to be offered in class on
Wednesday, September 3. Although the test is not timed,
most students in previous semesters have completed this test
in about 30 minutes. Once you have completed and turned in the benchmark
test, you may continue to work on the assigned computer lab activities.
This test will have ten problems on topics that are
prerequisite skills for calculus. These topics are reviewed in
Appendices A, B, and C, and in Chapter 1 of our text.
So if you are seriously working on the homework and class
activities during the first week of the semester, you will be
preparing for this benchmark test. You may not use a calculator while you
are working on the benchmark test.
- Prerequisite Algebra
-
Study pages A2 - A6 in Appendix A. You should be able to do
exercises #1 - 40 on pages A6 - A7.
- You should be able to:
- use the rules of exponents to simplify an algebraic expression;
- evaluate a given function, f(x), at particular values (such as at x = 5,
x = x+h, and x = g(x));
- solve an inequality involving one variable, and illustrate the
solution set on a number line;
- rewrite an expression involving the absolute value symbols without
using the absolute value symbols;
- solve an equation involving an expression with absolute value symbols;
- use the strategy of completing the square to transform an equation of
the form Ax^2 + Bx = C to an equation of the form
(ax + b)^2 = c.
- Coordinate Geometry
-
Study pages A7 - A12 in Appendix B. You should be able to do
exercises #1 - 45 and #53 - 54 on pages A16 - A17.
- You should know and be able to use:
- formulas for area of a square, rectangle, triangle, or circle;
- formulas for volume of a rectangular box, or ordinary cylinder;
- formulas for perimeter and circumference.
- You should be able to:
- locate points in 2-dimensional space using ordinary
rectangular coordinates;
- find the slope of a straight line from its graph, its algebraic
formula, or a set of points which lie on the line;
- identify whether two given lines are parallel or
perpendicular to each other;
- find the coordinates of the point of intersection of two lines
which are not parallel;
- find an equation of a line which is parallel or
perpendicular to a given line;
- find an equation of a line with a particular slope which goes through a
given point;
- find an equation of a line which passes through two given points;
- find the length and the midpoint of a line segment given its
endpoints:
- find the equation for a circle with a given center and given radius;
- sketch the graph of a straight line given its equation;
- sketch the graph of a straight line given its slope and one point;
- sketch the graph of a circle given its equation.
- Trigonometry
-
Study pages A18 - A22 (through formulas # 11), and pages A24 - A25
(graphs of trigonometric functions) in Appendix C.
You should be able to do
exercises #1 - 18, #29 - 40, #50, and #54 on pages A27 - A29.
- You should be familiar with basic right-triangle trigonometry:
- the six trigonometric relationships,
- the relationship between degrees and radians,
- the two standard triangles and the unit circle,
- the values of the trigonometric relationships of the angles in
the two standard triangles,
- the trigonometric relationships which can be verified in the unit
circle,
- the Pythagorean theorem, and the trigonometric versions of the Pythagorean
theorem.
Sample Questions for the Benchmark Test
In addition to the study problems listed above, the following sample
questions from calculus benchmarks in prior years
are offered as study questions for this first benchmark.
- Write the expression (sqrt (x^3)) in the form x^n.
- Write the expression for the cube root of (x^5)
in the form x^n.
- Write the expression 1 / (x^3) in the form x^n.
- Write the expression (x^4) / (x^7) in the form x^n.
What do fractional exponents mean? What do negative exponents
mean?
- If g(x) = 3 x^2, find an expression for g(-7).
- If g(x) = 3 x^2, find an expression for g(x + 5).
- If g(x) = 3 x^2, find an expression for g(cos(5)).
- If g(x) = 3 x^2, find an expression for g(cos(Pi/2)).
Although you are not expected to be able to evaluate
cos(5) by
hand, you should be able to evaluate the cosine or sine of angles of
0, Pi/6, Pi/4, Pi/3, Pi/2, Pi, 2 Pi radians,
and other special angles without using a calculator.
- Find the slope of a line which passes through the point (3, 5) and
goes through the origin.
- Find the equation of a line which passes through the point (3, 5) and
goes through the origin.
- Find the equation of a line which passes through the points (3, 5) and
(-3, 8)
- Find the equation of a line which passes through the points (3, 5) and
(3, -2)
- Find the slope of a line which passes through the points (3, 5) and
(3, -2)
- Find the equation of a line which passes through the point (3, 5) and
is perpendicular to the line y = -x.
- Is the line y = 5x + 4 [parallel to, perpendicular to, or neither parallel
nor perpendicular to] the line x + 5y = 20? How do you know?
- Is the line y = 5x + 4 [parallel to, perpendicular to, or neither parallel
nor perpendicular to] the line 5y - 10x = 20? How do you know?
- Find an angle whose tangent is -1.
- If the sin (t) is a, find an expression for the
sec (t).
- If the sin (t) is a, find an expression for the
cos (t) in terms of a.
- Evaluate sin (Pi/2).
- Evaluate cos (Pi/3).
- Evaluate tan (Pi/4).
- Evaluate cos (Pi).
- If an equilateral triangle has sides of length 3 units, what are the
lengths of its base and its altitude?
- If an equilateral triangle has sides of length 3 units, what is its
area?
- How many feet of fencing are needed to enclose a field which
is 76 yards wide and 50 yards long?
- A rectangular box has a volume of 600 cubic inches, and its height is
8 inches. If the length of the box is three times its width, what are the
dimensions (i.e., length, width, and height) of the box?
- A rectangular box has a volume of 600 cubic inches, and its height is
8 inches. If the length of the box is three times its width, what is the
surface area of the box?
- What is the volume of a cylinder if its diameter and its height
are both 3 inches?
- What is the total surface area of a cylinder (including its top
and sides) if its diameter and its height
are both 3 inches?
- Find the length of the line segment with endpoints at (2,3) and (-4,
5).
- What are the coordinates of the midpoint of the line segment with
endpoints (2,3) and (-4, 5)?
- Does the line through the points (2,3) and (-4, 5) pass through the
origin? How do you know?
-
To pass the benchmark test on the first attempt, you must get nine or
ten of ten problems completely correct; there will be no partial
credits. If you pass on the first attempt, your score will be recorded
as 100%. (So, clearly, there is some incentive for passing on the first
attempt!)
-
If you do not pass the benchmark test on your first attempt, you may
arrange to re-take the test up to two more times. You must
demonstrate that you have done some
additional study to prepare for the re-test, and then make an
appointment with me to take this test outside of class. Again, to
pass the benchmark test, you must get nine or ten problems correct with no
partial credits. You must
complete Benchmark #1 by Friday, September 20.
-
If you pass a benchmark test on the first or second re-test,
your score will be recorded as the average of your scores for all your
attempts. It is to your advantage to do as well as you can on your first attempt of this
benchmark.
-
If you have not passed this benchmark after two re-tests or by
Friday, September 20, your score will be recorded as 0%.
Answers to Sample Questions
- x^(3/2)
- x^(5/3)
- x^(-3)
- x^(-3)
- 147
- 3 (x+5)^2
- 3 (cos(5))^2
- 0
- 5/3
- y = (5/3) x
- y = (13 - x) / 2
- x = 3
- The slope is undefined (because its a vertical line).
- y = x + 2
- The lines are perpendicular, since the product of their slopes is -1.
- The lines are not parallel since their slopes are not equal; they are
not perpendicular since the product of their slopes is not -1.
- -Pi/4
- 1 / sqrt(1-a^2)
- sqrt(1-a^2)
- 0
- sqrt(3) / 2
- 1
- -1
- 3, 3 sqrt(3) / 2
- 9 sqrt(3) / 4
- 756 feet
- 15 inches, 5 inches, 8 inches
- 2 ( (15 * 5) + (15 * 8) + (5 * 8) ) square inches
- 27 Pi / 4
- (9 Pi / 2 + 9 Pi) square inches
- sqrt(45) = 3 sqrt(5)
- (-1, 4)
- 3/2 is not equal to -5/4, so the line through these points does not go
through the origin.
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This page was updated on August 23, 2003.