m223Review

m223Review - 1 MATH 223 REVIEW PROBLEMS 1 You are in a...

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1 MATH 223 REVIEW PROBLEMS 1. You are in a nicely heated cabin in the winter. Deciding that it’s too warm, you open a small window. Let be the temperature in the room, minutes after the window was opened, feet from the window. (a) Is an increasing or decreasing function of ? Explain. (b) Is an increasing or decreasing function of ? Explain. 2. Consider the following sets of points in space: A is the set of points whose distance from the -axis is . B is the set of points whose distance from the -plane is . C is the set of points whose distance from the -axis is equal to its distance to the -plane. Describe in words, write equations, and give a sketch for each of these sets of points. 3. Describe in words the intersection of the surfaces and , sketch it, and give simpliFed equations for it. 4. Each of the graphs in ±igure 1 shows the quantity that can be sold of a certain product (Product ), as a function of its price and the price of another product (Product ). (a) (b) (c) Figure 1 Choose which graph Fts best and explain why if: (a) Product is ²ashlights and Product is batteries. (b) Product is domestic cars and Product is imported cars. 5. Match each of the following functions in (a)-(f), given by a formula, to the corresponding table of numerical values, graph and/or contour maps (i)-(xi). Some formulas may match NONE or MORE THAN ONE of the representations (i)-(ix). (a) (b) (c) (d) (e) (f)
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2 (i) Table 1 (ii) Table 2 (iii) Table 3 (iv) Figure 2 (v) Figure 3 (vi) Figure 4 (vii) Figure 5 (viii) Figure 6 (ix) Figure 7
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3 6. Draw a possible contour diagram for the function whose graph is shown below. Label your contours with reasonable -values. Figure 8 7. Consider the function . (a) Plot the level curves of the function for . Label them clearly. (b) Imagine the surface whose height above any point is given by . Suppose you are standing on the surface at the point where , . (i) What is your height? (ii) If you start to move on the surface parallel to the -axis in the direction of increasing , does your height increase or decrease? Justify your answer using your contour map. (iii) Does your height increase or decrease if you start to move on the surface parallel to the -axis in the direction of increasing ? Justify your answer using your contour map. 8. Figure 9 below shows the level curves 1 of the temperature in degrees Celsius as a function of hours and depth in centimeters beneath the surface of the ground at O’Neill, Nebraska from midnight one day to midnight the next. Figure 9 (a) Approximately what time did the sun rise? When do you think the sun is directly overhead? Explain your reasoning. (b) Sketch graphs of the temperature as a function of time at centimeters and at centimeters below the surface. (c) Sketch a graph of the temperature as a function of the distance at noon.
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m223Review - 1 MATH 223 REVIEW PROBLEMS 1 You are in a...

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