This benchmark test will contribute 5% to your final course grade.
You may take this benchmark by appointment with the instructor any time between
December 2 and December 9. This test will have ten problems on basic
computational skills and strategies that we have
been using throughout this course. You will not be allowed to use a
calculator or computer as you work on the benchmark test.
- To pass the benchmark test, you must get nine or
ten of the ten problems completely correct; there will be no partial
credits. If you pass on the first attempt, your score will be recorded
as 100%.
- If you do not pass the benchmark test on your first attempt, you may
demonstrate that you have done some additional practice, and make an
appointment with the instructor to try the test again once or twice.
If you pass the test on the first or second retest,
your score will be recorded as the average of your scores.
- If you do not pass the benchmark test by December 9,
your score will be recorded as 0%.
- Sequences and Discrete Dynamical Systems
-
You should be able to:
- Write out the first several terms for a sequence which is given as a
discrete dynamical system.
- Write an expression for the (n+1)st term of a sequence as a function
of the previous terms for a given sequence.
- Find an equilibrium value (if one exists) for a given dynamical
system, and classify the system as stable or unstable.
- Difference Quotient
-
You should be able to set up and simplify a difference quotient for a
given function.
- Derivatives
-
You should be able to:
- Calculate derivatives for polynomials, and for the six
trigonometric functions.
- Use the product rule, the
quotient rule, and the chain rule to calculate derivatives for simple
combinations of functions.
- Calculate a specified partial derivative for a given function of more
than one variable.
- Antiderivatives and Definite Integrals
-
You should be able to:
- Use the power rule to integrate an expression of
the form x^n, both for n /= -1 and for n = -1.
- Calculate antiderivatives and definite integrals
for polynomials,
and for the derivatives of the six trigonometric functions.
- Use simple substitutions to calculate an
antiderivative. (Essentially, this means that you can find the
antiderivative when the derivative was calculated using the chain rule.)
- Evaluate an antiderivative or definite integral of an expression
of the form e^u du.
- Calculate the antiderivative or definite integral for
an expression of the form df(x)/f(x) dx.
- Express a rational function, p(t) / q(t), as a sum of
partial fractions.
- Use a partial fraction decomposition to simplify an expression of the
form p(t) / ( (x-a)(x-b)(x-c) ) to an expression that is easier
to integrate. (And then you should be able to integrate the expression.)
- Solve an elementary differential equation.
- Algebra and Calculus
-
You should be able to:
- Transform an expression of the form y = A x^n to a linear form.
- Solve ln (y) = x and e^y = x for y.
- Elementary Geometry and Trigonometry
-
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 of simple geometric figures.
You should be familiar with basic unit-circle 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.
Study Questions for the Benchmark Test
- Write out the first five terms for the sequence a(n+1) = 2 *
a(n) + 6, a(0) = -3. Is this system in equilibrium? Why or why not?
- Consider the sequence [1, 1/2, 2/3, 3/4, ...].
- Find the next term in the sequence.
- Construct an expression giving the general (or n-th) term of the
sequence.
- Consider the sequence whose n-th term is: n^3 / n!
Write out the first four terms of this sequence.
- Set up a dynamical system to
represent this sequence: [160, 200, 250, 312.5, ... ].
- Find an equilibrium value for a(n+1) = 0.9 * a(n), and determine
whether this equilibrium value is stable or unstable.
- If the sin (t) = 3/4, find an expression for the sec (t)
(without using a calculator).
- What is the value of the cosine of 30-degrees?
- What is the tan(Pi/4)?
- Find an angle bigger than Pi/2 whose tangent is 1.
- Arrange these values in increasing order of magnitude (from smallest to
largest): sin(Pi/3), cos(Pi/3), tan(Pi/3).
- What is the area of a circle whose diameter is 2*Pi inches?
- What is the diameter of a circle whose area is 25 Pi square inches?
- What is the surface area of a box whose dimensions are 4 cm by 5 cm by
6 cm?
- What is the volume of a box whose dimensions are 4 cm by 5 cm by 6 cm?
- What is the surface area (including top and bottom)
of a tin can with radius 3 inches and height 4 inches?
- What is the volume of a tin can with radius 3 inches and height 4
inches?
- Set up a difference quotient for g(x) = 3x^2. Show that this can be
simplified to 6x + h.
- Set up a difference quotient for g(x) = 3/x. Show that this can be
simplified to -3 / [x (x+h)].
- Set up a difference quotient for g(x) = sqrt(x+3). Show that this
can be simplified to 1 / [sqrt(x+h+3) + sqrt(x+3)].
- Set up and simplify a difference quotient for y = sqrt(x).
"Prove" the basic derivative rule for sqrt(x).
- Calculate the derivative of each of the following:
- y = sin(x) + 3x^2
- y = sin(x) * 3x^2
- y = sin(x) / (3x^2)
- y = sin(3x^2)
- y = (sin(3x^2)) ^ 3
- x^2 + 3x - 1/x
- [x^2 + 3x] / tan(x)
- 3 e^x + ln (5x)
- x^2 cos(x) sin(3x)
- Evaluate the derivative of x^2 + sin(x) at x = pi/4.
- Calculate each of the following antiderivatives:
- int (sin(x) + 3x^2, x)
- int (5 cos(3x), x)
- int ((sin(2x) / cos(2x)), x)
- int (x * sin(3x^2), x)
- int (x * sin(x), x)
- int (ln(x), x)
- int (e^x * sin(x), x)
- int (sec(2x) * tan(2x), x)
- int (x * (sec(3x^2))^2, x)
- int (x^2 * sin(x^3) * sec(x^3), x)
- int ((sec(x))^2, x)
- int ( (5x+7) / (x^2 + 3x +2), x)
- int (2 / (x^2 - 1), x)
- int ((x^2 + 3x -1) / (x^3 - x^2 - 2x), x)
- Evaluate the definite integral of (cos (3t) dt) over
the interval from -Pi to 3Pi/2.
- Find df/dt if f(s,t) = 3 * t^2 + sqrt (s) - 5/t + s*t.
- Find an antiderivative of (cos (3t) dt)/ sin (3t).
- Solve ln(3y) = 2x for y.
- Solve exp(3y) = 2x + 5 for y.
- Solve ln(3y) - e^2 = x^2 + x for y.
- Find y = f(t) if dy/dt = k y, where k is a constant.
- Find y = f(t) if dy/dt = 10 y (10-y).
- Transform the equation y = a x^2 to a linear form. If you
were to plot this linear form on a pair of coordinate axes, what variable (or
values) would you put on the horzontal axis, and what variable (or values)
would you represent on the vertical axis. What would the slope and the intercept
of the transformed equation represent?
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