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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 ht 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.