Download MATH 12001 Learning Outcomes Kent Core Learning Outcome

MATH 12001 Learning Outcomes
I. Kent Core Learning Outcome
Acquire critical thinking and problem-solving
skills.
Understand basic concepts of the academic
discipline.
Apply principles of effective written and oral
communications.
II. State TMM-001 and
III. Course Learning Outcome
IV.
Sample Assessment Items
1. Analyze functions. Routine analysis includes 1. A 3 m x 1.5 m piece of plywood is being used to
build an open-top toy chest. The chest is formed
discussion of domain, range, zeros, general
by making equal-sized square cutouts from four
function behavior (increasing, decreasing,
corners of the plywood. After these squares are
extrema, etc.).
discarded, these pieces are “folded up” and
In addition to showing procedural fluency, the
secured to create the open top toy chest.
student can articulate reasons for choosing a
Neglect the thickness of the wood when
particular process, recognize function families
answering these questions.
a) If the variable x represents the length of the
and anticipate behavior, and explain the
sides of the square cutouts in meters,
implementation of a process
define a function f that expresses the total
volume of the box in terms of x, the length
of the side of the square cutouts, x.
b) Name the domain of the function defined in
a).
c) Given the following graph of the f, name the
intervals of x for which the volume is
increasing.
d) Name the dimensions of the box for which
the volume is a maximum.
2.
Find the domain of the function, g, given by
g x 
x 7
. Write your answer using interval
x 8
Notation and explain why both restrictions are
Necessary.
18
3. Name the16domain and range of the function
whose graph is given.
14
12
10
8
6
4
2
-35
-30
-25
-20
-15
-10
-5
5
-2
-4
-6
-8
-10
-12
-14
10
15
20
25
Understand basic concepts of the academic
discipline.
2. Convert between different representations of
a function.
Perform operations with functions including
addition, subtraction, multiplication, division,
composition, and inversion.
4. Determine the specific growth or decay factor,
percent change, initial value, and function for
the table modeled by an exponential function.
x
1
4
7
10
260
278.2
297.674 318.511
g x
a)
b)
c)
d)
e)
f)
3-unit growth factor
6-unit percent change
1-unit growth factor
¼ unit growth (decay) factor
Initial value
Function
6. Use the graphs of f and g to evaluate g  f  3   .
8
(3,7)
6
g
4
(7,3)
2
(-3,2)
-15
-10
f
-5
5
10
-2
(3,-3)
-4
7. Evaluate
f 3  g  3
-6
8. Let ht   3t  1 and let k  s   4 s2 . Find a function
rule for k  h  n   .
9. Let f  x  
Acquire critical thinking and problem-solving
skills.
Understand basic concepts of the academic
discipline.
Strengthen quantitative reasoning skills.
Apply principles of effective written and oral
communications.
3. Recognize function families as they appear in
equations and inequalities and choose an
appropriate solution methodology for a
particular equation or inequality and can
communicate reasons for that choice.
Demonstrate an understanding of the
correspondence between the solution to an
equation, the zero of a function, and the point
of intersection of two curves.
7x  3
, find f 1  y 
4
10. When a new piece of technology is invented
and sold, its value decreases over time because
it wears down and because even newer, better
technology is developed to replace it. Suppose
two electronic devices have the same resale
value right now and that their resale values are
expected to change according the following
patterns:
 Product #1: iTech Device. The current
resale value is $300. The resale value is
expected to decrease by 30% per year for
the next several years.
15

Product #2: Dynasystems Device. The
current resale value is $300. The resale
value is expected to decrease $45 per year
for the next several years.
a) Write a function for each of the above
scenarios.
b) Predict when each item will have 0 value.
Understand basic concepts of the academic
discipline.
4. Use correct, consistent, and coherent notation 11. Determine the difference quotient (i.e. the
throughout the solution process to a given
average rate of change over an input of length h)
equation or inequality.
for the function f given by f  x   6 x2  7x  11 .
Distinguish between exact and approximate
solutions and which methods results in which 12. Without a calculator, approximate the solutions
kind of solutions.
to the following equations:
Solve for one variable in terms of another.
Solve systems of equations using substitution
a) 2x  10 b) 3x  10
c) 17x  10
or elimination.
13. Using logarithmic notation, represent exact
solutions to the above equations.
Acquire critical thinking and problem-solving
skills.
Strengthen quantitative reasoning skills.
Apply principles of effective written and oral
communications.
Broaden their imagination and develop their
creativity.
5. Purposefully create equivalencies and indicate
24 x 3  60 x
14. Simplify as much as possible:
when they are valid.
4 x
Recognize opportunities to create
15. Solve for x:
equivalencies in order to simplify workflow.
a) log2 2  x   log 7  3
 
b) ln 3x2  ln 5x   ln x  9 
16. Identify the x-intercepts, y-intercepts, horizontal
asymptotes, vertical asymptotes, and the
domain, then sketch a graph of the function f,
given by f  x  
x2  2x  3
x2  1
Cultivate their natural curiosity and begin a
lifelong pursuit of knowledge.
Integrate their major studies into the broader
context of a liberal education.
Broaden their imagination and develop their
creativity.
Strengthen quantitative reasoning skills.
6. Interpret the function correspondence and
behavior of a given model in terms of the
context of the model.
Create linear models from data and interpret
slope as rate of change.
17. Each function below defines the population for a
city in terms of the time t years since the city
was established. Write a sentence that
describes the city’s initial population and the
growth pattern.
a) f t   2000 1.24 t
b) g t   1500  20t
c) h t   4000  0.68 t
d) k t   2500  40t
e) f t   1500 1.4 
t
2
18. Nick is considering joining a weight-loss club
that provides meals and support for peoples
who want to lose weight. Based on an initial
consultation with a weight-loss advisor, Nick
charted his potential weight loss based on the
advisor’s estimates of his expected weekly
weight loss.
# Weeks since joining
3
6
8
12
13
Projected weight (lbs)
277
266.5
259.5
245.5
242
a) Determine if the relationship above is linear.
b) If so, write a function f relating Nick’s projected
weight in term of the number of weeks since he
joined the club.
c) Name and interpret the slope f.
Acquire critical thinking and problem-solving
skills.
Strengthen quantitative reasoning skills.
Apply principles of effective written and oral
communications.
7. Determine parameters of a model given the
form of the model and data.
Determine a reasonable applied domain for
the model as well as articulate the limitations
of the model.
19. An engagement ring is thrown upward from the
second story of a house (22 feet above the
ground). The ring reaches its maximum height
above the ground 0.36 seconds after it is thrown
and contacts the ground 1.02 seconds after it
was thrown. Define a quadratic function, g, that
gives the height of the ring above the ground (in
feet) in terms of the number of seconds elapsed
since the ring was thrown, t. What would be a
reasonable domain for your function?
20. Define the function formula that generates a
parabola that has horizontal intercepts x  3 and
x  5 , and passes through the point 2,3
21. The population of Canada in 1990 was
27,512,000 and in 2000 it was 30,689,000.
Assume that Canada’s population increased
exponentially over this time.
a. Determine the initial value, the 10 year
growth factor, the 10-year percent change
of Canada’s population
b. Define a function f that defines the
population of Canada in terms of the
number of decades (10-year periods) since
1990.
c. Assume Canada’s growth rate remains
consistent, predict the approximate
population of Canada in 2013.
d. Define a function g defines the population
of Canada in terms of the number
22. As Pat was driving his hybrid cruiser across
Kansas with his cruise control on , his gas gauge
broke. At the moment the gauge broke, he had
15 gallons of gas in the car’s gas tank and his
gas mileage was 41 miles per gallon (assume
that he maintains this gas mileage by leaving
cruise control on). Pat needs to keep track of
how much gas is left in his gas tank.
a. How many gallons does Pat have left after
he has driven 84 miles? 150 miles?
b. Define a function f to determine the
number of gallons left in the tank f(x) in
terms of the number of miles driven.
c. What is the domain of f?
d. What is the range of f?
e. Explain what
x
represents in the context
42
of the problem
f.
Explain what the expression 15 
x
42
represents in the context of the problem.
g. What are the max and min values that f(x)
can assume in the context of this situation?
Explain.
h. Construct a graph of f on the given axes.
Explain what the graph of f conveys about
how the number of miles Pat has driven
since leaving home x and the number of
gallons of gas left in Pat’s tank f(x) change
together.
i. What does the point (0,15) represent in the
context of this situation?
Acquire critical thinking and problem-solving
skills.
Develop competencies and values vital to
responsible uses of information and
technology.
Acquire critical thinking and problem-solving
skills.
Strengthen quantitative reasoning skills.
Understand basic concepts of the academic
discipline.
Apply principles of effective written and oral
communications.
8. Anticipate the output from a graphing utility
23. A biologist counted 426 bees in a bee colony.
and make adjustments, as needed, in order to
His tracking of the colony revealed that the
efficiently use the technology to solve a
number of bees increased by 4% each month
problem.
over the next year.
Use technology to verify solutions to
a) Use your graphing calculator to estimate
equations and inequalities which are difficult
how many months pass before the number of
to obtain algebraically and know the
bees reached 500.
difference between approximate and exact
b) Solve algebraically and compare your
solutions.
answer with that obtained in a) above.
Use technology and algebra in concert to
locate and identify exact solutions.
9. Recognize when a result is applicable and use 24. Use the Rational Zero Theorem to help identify
the result to make sound logical conclusions
approximate solutions, then uses the Factor
and provide counter-examples to conjectures.
Theorem to verify the zeros.
1. Represent trigonometric and inverse
trigonometric functions verbally, numerically,
graphically and algebraically; define the six
trigonometric functions in terms of right triangles
and the unit circle.
25. An ice cream shop finds that its weekly profit P
(measured in dollars) as a function of the price x
(measured in dollars) it charges for ice cream
cone is given by the function k, defined by
k  x   125x2  670x  125 where P  k  x 
a) Determine the maximum weekly profit and
the price of an ice cream cone that produces
that maximum profit
b) Determine what price the ice cream shop
needs to charge in order to break even.
1. Find sin 𝜃, where ⁡𝜃 is shown below:
2. Suppose (-3/5, 4/5) is on the terminal side of α.
Find the tangent of α.
Understand basic concepts of the academic
discipline.
Strengthen quantitative reasoning skills.
2. Perform transformations of trigonometric and
inverse trigonometric functions – translations,
reflections, stretching/shrinking (amplitude,
period, phase shift)
3. Find the amplitude, period, and phase shift of of
Acquire critical thinking and problem-solving
skills.
Understand basic concepts of the academic
discipline.
Strengthen quantitative reasoning skills.
3. Analyze the algebraic structure and graph of
trigonometric and inverse trigonometric functions
to determine intercepts, domain, range, intervals
of increase/decrease, asymptotes, whether a
function is one-to-one, whether a graph is
even/odd, and given the graph of a function to
determine possible algebraic definitions.
4. Starting with the definition of the tangent
function, find its zeros and vertical asymptotes and
describe the behavior of the graph near the
asymptotes.
Acquire critical thinking and problem-solving
skills.
Strengthen quantitative reasoning skills.
Apply principles of effective written and oral
communications.
Acquire critical thinking and problem-solving
skills.
Strengthen quantitative reasoning skills.
Apply principles of effective written and oral
communications.
Broaden their imagination and develop their
creativity.
Cultivate their natural curiosity and begin a
lifelong pursuit of knowledge.
4. Use trigonometric and inverse trigonometric
functions to model a variety of real-world
problem-solving applications.
5. A steel ring of radius R is positioned with its
center at the origin of an xy-plane and is cut at the
two points along the vertical line 𝑥 = 𝑎, with 0 <
𝑎 < 𝑅. Write the length of the smaller piece as a
function of 𝑎.
6. Find the exact real number value or radian
measure or state “undefined”.
5. Solve a variety of trigonometric and inverse
trigonometric equations, including those requiring
the use of fundamental trigonometric identities, in
degrees and radians for both special and nonspecial angles. Solve application problems that
involve such equations.
𝜋
𝑆(𝑥) = 3 cos (2𝑥 − 3 ) + 2 and sketch its graph.

 5  
 
 6 
11 

6 
a) arcsin  sin 




b) sin  cos 1
7. The height of a cell phone tower is 100 meters,
and it casts a shadow of length 120 meters. Draw a
right triangle that gives a visual representation of
the situation and label the known and unknown
Understand basic concepts of the academic
discipline.
Strengthen quantitative reasoning skills.
6. Express angles in both degree and radian
measure.
quantities. Then find the angle of elevation of the
sun to the nearest tenth of a degree.
8. Convert 260 to radians. Leave the simplified
answer in terms of  .
9. In the circle shown, find the measure of the
central angle in radians and then convert it to
degrees.
s  r
s = 3.5cm

r = 10 cm
Acquire critical thinking and problem-solving
skills.
Strengthen quantitative reasoning skills.
7. Solve right and oblique triangles in degrees and
radians for both special and non-special angles,
and solve application problems that involve right
and oblique triangles.
10. A person sees a hot air balloon in the distance.
The angle of elevation from the ground to the
balloon is 35. If the balloon is 480 feet in the air,
determine the distance from the balloon to the
person to the nearest foot.
11. On a small lake, Chuck plans to start at point A
and swim to point B. He gets drawn off course and
ends up swimming 458 feet to point C. He notices
his mistake, turns 80 and swims 261 feet to point
B. How far would it be if he swam directly from
point A to point B?
B
80
C
A
Acquire critical thinking and problem-solving
skills.
Apply principles of effective written and oral
communications.
Broaden their imagination and develop their
creativity.
Acquire critical thinking and problem-solving
skills.
Strengthen quantitative reasoning skills.
8. Verify trigonometric identities by algebraically
manipulating trigonometric expressions using
fundamental trigonometric identities, including
the Pythagorean, sum and difference of angles,
double-angle and half-angle identities
12. Verify that the given equation is an identity.
9. Represent vectors graphically in both
rectangular and polar coordinates and understand
the conceptual and notational difference between
a vector and a point in the plane.
13. a) Draw a sketch and find the b) magnitude and
the
sin  x  y   sin  x  y 
 2sin x cos y
c) component form of vector MN ;
M  2,3 , N  4, 6
14. Determine the
a) magnitude and the
b) direction angle θ of w  2, 6
Understand basic concepts of the academic
discipline.
10. Perform basic vector operations both
graphically and algebraically – addition,
subtraction, and scalar multiplication.
15. Using the vectors t  3i  j and u  5i  4 j ,
simplify the expression and write your answer in
terms of i and j .
2u  5t
Acquire critical thinking and problem-solving
skills.
Strengthen quantitative reasoning skills.
11. Solve application problems using vectors.
16. A crate weighing 250 lbs. is suspended by two
cables as shown in the figure below. Find the
tension in each cable.