Calculus is Cool. Math 160 Calculus for Physical Scientists I Exam 1 September 18, 2014, 5:00-6:50 pm. NAME: Instructor: Time your class meets:

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1 NAME: Instructor: Time your class meets: Math 160 Calculus for Physical Scientists I Exam 1 September 18, 2014, 5:00-6:50 pm How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? -Albert Einstein 1. Turn off your cell phone and other devices (except your calculator). 2. Write your name on every page of the exam. Write your instructor s name on the cover sheet. 3. You may use a calculator on this exam. You must provide your own calculator; you may not use a laptop computer or smart phone. 4. No notes or other references, including calculator manuals or notes stored in calculator memory, may be used during this exam. 5. Use the back of the facing pages for scratch work and for extra space for solutions. Indicate clearly when you wish to have work on a facing page read as part of a solution to a problem. HONOR PLEDGE I have not given, received, or used any unauthorized assistance on this exam. Furthermore, I agree that I will not share any information about the questions on this exam with any other student before graded exams are returned. (Signature) (Date) Please do not write in this space (15pts) 6. (12pts) 7. (15pts) 8. (3pts) 9. (12pts) 10. (15pts) 11. (12pts) Calculus is Cool 12. (16pts) TOTAL

2 Multiple Choice for #1-5 (15pts - 3pts each). Use the function h(x) to answer the following multiple choice questions. Circle only one answer for each problem. cos(x), x < 0 h(x) = 2, x = 0 cos(x) + 1, x > 0 1. lim x 0 h(x) = (a) -2 (b) 0 (c) 1 (d) 2 (e) π (f) Does not exist 2. lim x 0 h(x) = (a) -2 (b) 0 (c) 1 (d) 2 (e) π (f) Does not exist 3. lim x 0 + h(x) = (a) -2 (b) 0 (c) 1 (d) 2 (e) π (f) Does not exist 4. h(0) = (a) -2 (b) 0 (c) 1 (d) 2 (e) π (f) Does not exist 5. Circle the graph that represents the graph of h(x). (a) (d) (b) (e) (f) None of the above. (c)

3 8t 6. (12pts) Who correctly evaluated lim t 0 3 sin(t) t? Below are three different student solutions to the limit. Read over each solution from each student and determine who correctly evaluated the limit and why. Taylor s Solution: sin(0) 0 = = 0 0 The limit does not exist. Jimminy s Solution: 8t 3 sin(t) t = 8t ( ) = t 3 sin(t) 1 t 8 3 sin(t) 1 = = 8 2 = 4 t Margo s Solution: lim t 0 8t 3 sin(t) t = lim t 0 8t ( t 3 sin(t) 1 t 8 ) = lim t 0 3 sin(t) 1 = = 8 2 = 4 t Taylor correctly / incorrectly (circle one) evaluated the limit. If you circled incorrectly, state why Taylor s evaluation is incorrect. Jimminy correctly / incorrectly (circle one) evaluated the limit. If you circled incorrectly, state why Jimminy s evaluation is incorrect. Margo correctly / incorrectly (circle one) evaluated the limit. If you circled incorrectly, state why Margo s evaluation is incorrect.

4 7. (15pts) Below is a graph of the position of a mass oscillating up and down on a spring over time. The function that gives the height above the ground, h, of the spring at time, t, is given by h(t) = cos(πt) + 2 where time is measured in seconds, sec, and height is measured in centimeters, cm. (a) (2pts) At what times during the first 4 seconds is the mass at its highest point? (b) (9pts) Find the average speed of the spring on the following time intervals. Round values to 4 decimal places. i. [2.5, 2.51] ii. [2.5, 2.501] iii. [2.5, ] (c) (4pts) Based on your answers above, how fast would you expect the mass to be moving at exactly t=2.5? Explain.

5 8. (3pts) From the mathematical definition of continuity, we know a function f(x) is continuous at an interior point x = c of its domain if and only if (circle one). (a) There are no holes, vertical asymptotes, or jumps at x = c. (b) lim x c f(x) exists and is a real number. (c) lim x c f(x) = f(c). (d) f(c) exists. 9. (12pts) Consider the function V (x) given by x 2b, x < 0 V (x) = a + 1, x = 0 x 2 + b, x > 0 where a and b are constants. (a) Find the following. Simplify your results and write your answers in terms of a and b. i. (2pts) V (0) = ii. (3pts) lim V (x) = x 0 iii. (3pts) lim V (x) = x 0 + (b) (4pts) Find a, b such that V (x) is continuous at every point in its domain. Write your answers in the blanks below. Be sure to provide supporting work. a = b =

6 10. (15pts) You are grinding engine cylinders for a company. You receive an order for cylinders that requires a circular cross-sectional area of 9 in 2. (a) In the blank below, write the function that relates the circular cross-sectional area, A, and the cylinder diameter, d. A(d) = (b) What is the perfect diameter? i.e. What diameter will result in a circular cross-sectional area of 9 in 2? Your answer should be written to 4 decimal places. d 0 = (c) The circular cross-sectional area must be within 0.01 in 2 of 9 in 2. Algebraically determine the interval around d 0 will ensure that corresponding output values are within 0.01 in 2 of 9 in 2. < d 0 < (d) How much can you deviate from the perfect diameter and still have the circular cross-sectional area still be within 0.01 in 2 of 9 in 2? (CIRCLE ALL CORRECT RESPONSES) i in ii in iii in iv in v. None of the above (e) Below is the portion of the graph of A(d) relevant for this problem. Using the graph, label the following: i. d = d 0 ii. A(d) = 9 in 2 iii. The interval of ±0.01 in 2 around A(d) = 9 in 2. iv. The interval you found in part (c).

7 11. (12pts) Sketch the graph of a function that has the following properties: lim x F (x) = F ( 5) = 5 lim F (x) = 5 x 5 lim F (x) = 3 x 5 + lim F (x) = x 0 F (5) = 3 lim F (x) = 3 x 5 lim F (x) = 5 x 5 + lim x F (x) = 7 F (x) has a vertical asymptote at. F (x) has as a horizontal asymptote at.

8 12. (16pts - 4pts each) Indicate whether each of the following statements is True or False. If the statement is true, explain how you know it s true. If it is false, give a counterexample and explain why it is a counterexample. (A counterexample is an example of a function for which the if part of the statement is true, but the then part is false.) A graph with an explanation can be used as a counterexample. If you use a term or phrase such as continuity or average rate of change, be sure to state the definition of the term or phrase that you used. (a) If f(π) = π and f(x) is continuous, then lim f(x) = π. x π (b) If the function f(x) has x = 2 as a vertical asymptote, then f(2) cannot exist. (c) If lim g(x) = 3, then lim g(x) = 3. x x (d) If lim x 17 h(x) does not exist, then h(17) cannot exist.

9 NAME: Instructor: Time your class meets: Math 160 Calculus for Physical Scientists I Exam 2 October 16, 2014, 5:00-6:50 pm How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? -Albert Einstein 1. Turn off your cell phone and other devices (except your calculator). 2. Write your name on every page of the exam. Write your instructor s name on the cover sheet. 3. You may use a calculator on this exam. You must provide your own calculator; you may not use a laptop computer or smart phone. 4. No notes or other references, including calculator manuals or notes stored in calculator memory, may be used during this exam. 5. Use the back of the facing pages for scratch work and for extra space for solutions. Indicate clearly when you wish to have work on a facing page read as part of a solution to a problem. HONOR PLEDGE I have not given, received, or used any unauthorized assistance on this exam. Furthermore, I agree that I will not share any information about the questions on this exam with any other student before graded exams are returned. (Signature) (Date) Please do not write in this space (17pts) 4. (7pts) 5. (15pts) 6. (12pts) 7. (15pts) 8. (12pts) 9. (8pts) 10. (14pts) TOTAL Calculus is awesome!

10 Multiple Choice for #1-4. Circle only one answer for each problem unless it indicates otherwise. 1. (3pts) At the point (0, 0), the graph of f(x) = x (a) has y = 0 as a tangent line. (b) has infinitely many tangent lines. (c) has no tangent line. (d) has y = x and y = x as both of its tangent lines. (e) none of the above 2. (8pts) Below is the graph of a function that changes with respect to time. Which of the following statements are accurately modeled by the graph? (CIRCLE ALL CORRECT RESPONSES) For each response that you circle, fill in the blank with the function represented by the graph (position, velocity, or acceleration). (a) Olga climbs to the top of a mountain but quickly descends to a shelter halfway down when she sees a thunderstorm on the horizon. (b) Tatiana is jumping on a trampoline until her foot slips and she falls to the ground. (c) Alexie accelerates from a stop sign before reaching a school zone and needing to slow down to a legal speed. (d) Maria s plane reaches cruising altitude and stays there for the rest of the flight. (e) Anastasia is proud of her efforts during the Fort Collins 10K Race, since she never had to slow down. [Note: A 10K race is a distance of 10 kilometers (6.2 miles)]

11 3. (6pts) Which of the following statements are true? (CIRCLE ALL CORRECT RESPONSES) (a) If f(x) is continuous at x = 5, then f (5) exists. (b) If f(x) is continuous at x = 5, then f (5) does not exist. (c) If f(x) is continuous at x = 5, then f (5) may or may not exist. (d) If f(x) is not continuous at x = 5, then f (5) does not exist. (e) If f(x) is not continuous at x = 5, then f (5) may or may not exist. (f) If f (5) does not exist, then f(x) is not continuous at x = 5. (g) If f (5) exists, then f(x) is continuous at x = Circle the correct response and then explain your answer below. (2pts) Two racers start a race at exactly the same moment and finish at exactly the same moment (they tied at the finish). Which of the following statements must be true. Explain how you know. (a) At some point during the race, the two racers were not tied. (b) The racer s speed at the end of the race was exactly the same. (c) The racers must have had the same speed at exactly the same time at some point in the race. (d) The racers had to have the same speed at some moment, but not necessarily at exactly the same time. Explain (5pts):

12 5. (15pts - 5pts each) Use the given information in the table to find the following derivatives: x f(x) f (x) g(x) g (x) π (a) d dx ( ) f(x) x 2 x=1 (b) d dx (f(x) g(x)) x= 2 (c) d dx (f(g(x))) x=1

13 6. (12pts - 4pts each) Indicate whether each of the following statements is True or False. If the statement is true, explain how you know it s true. If it is false, give a counterexample and explain why it is a counterexample. (A counterexample is an example of a function for which the if part of the statement is true, but the then part is false.) A graph with an explanation can be used as a counterexample. (a) If f(x) is defined on the interval [ 3, 3], then f(x) must have a maximum on [ 3, 3]. (b) Given that f(1) = f(3) = 0 and f (2) = 0, then f(x) must be continuous on the interval [1, 3] (c) Two different functions, f(x) and g(x), cannot have the same derivative functions unless both f(x) and g(x) are linear functions with the same slope.

14 7. (15pts) Use f(x) = x + x 2 to answer the following questions (a) (5pts) Sketch an accurate graph of f(x) in the axes below. [An accurate graph shows the function s domain, has the correct shape, and key points on the graph have the correct coordinates.] (b) (10pts) Using the definition of the derivative (as a limit), determine if f (0) exists. f (0) does / does not (CIRCLE ONE) exist.

15 8. (12pts) Below are the graphs of a position function s(t), a velocity function, v(t), and an acceleration function a(t) with respect to time, t. Which graph is position? Which graph is velocity? Which graph is acceleration? Give reasons for your answers in sentences. Your explanation should include a discussion of slope with regard to each graph. Graph 1 = position s(t), velocity v(t), acceleration a(t) (CIRCLE ONE) Graph 2 = position s(t), velocity v(t), acceleration a(t) (CIRCLE ONE) Graph 3 = position s(t), velocity v(t), acceleration a(t) (CIRCLE ONE)

16 9. (8pts) In the axes provided, sketch the graph of a function that has the following properties. You MUST label each property of your graph with the corresponding letter. (a) At point x = 0, f (0) does not exist, but lim f(x) = f(0) x 0 (b) lim f(x) = 0 x 2 + (c) (d) lim f(x) = x 2 lim f(x) = 1 x (e) lim x f(x) = 0 (f) A local maximum at x = 1

17 10. Use the curve of sin(πx) + cos(πy) = sin(x) below to answer the following questions: LEAVE ALL ANSWERS IN EXACT FORM. DO NOT USE DECIMALS. (a) (2pts) Draw the line tangent to the curve at the point (b) (8pts) Find dy dx ( 0, 1 ). 2 using implicit differentiation. Show all work. (c) (4pts) Find the equation of the tangent line you drew in (a). ( (i.e. find the equation of the line tangent to the curve at the point 0, 1 ) ). 2 LEAVE ALL ANSWERS IN EXACT FORM. DO NOT USE DECIMALS.

18 NAME: Instructor: Time your class meets: Math 160 Calculus for Physical Scientists I Exam 3 November 13, 2014, 5:00-6:50 pm How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? -Albert Einstein 1. Turn off your cell phone and other devices (except your calculator). 2. Write your name on every page of the exam. Write your instructor s name on the cover sheet. 3. You may use a calculator on this exam. You must provide your own calculator; you may not use a laptop computer or smart phone. 4. No notes or other references, including calculator manuals or notes stored in calculator memory, may be used during this exam. 5. Use the back of the facing pages for scratch work and for extra space for solutions. Indicate clearly when you wish to have work on a facing page read as part of a solution to a problem. HONOR PLEDGE I have not given, received, or used any unauthorized assistance on this exam. Furthermore, I agree that I will not share any information about the questions on this exam with any other student before graded exams are returned. (Signature) (Date) Please do not write in this space (12pts) 4. (14pts) 5. (12pts) 6. (15pts) 7. (20pts) (12pts) 11. (15pts) TOTAL rutabaga 2 = abaga

19 Multiple Choice for #1-4. Circle only one answer for each problem unless it indicates otherwise. 1. Let f(x) be a differentiable function on a closed interval where x = c is one of the endpoints of the interval and f (c) > 0. (a) f could have an absolute maximum or an absolute minimum at x = c. (b) f cannot have an absolute maximum at x = c. (c) f must have an absolute minimum at x = c. (d) f must have an absolute maximum at x = c. 2. If f is an antiderivative of g, and g is an antiderivative of h, then (a) h is an antiderivative of f. (b) h is the second derivative of f. (c) h is the derivative of f. 3. Let f(x) = { x 2, x 3 7x 12, x > 3 Which of the following statements is true? (a) (b) (c) (d) f(x) dx > 0. f(x) dx < 0. f(x) dx = 0. f(x) dx is undefined.

20 4. Water is flowing into a boat through a hole at the bottom at a rate of r(t). Water is flowing in at increasing rates for the first 10 minutes and then at decreasing rates after the first 10 minutes. You do not know the function, r(t), but you do have values of r(t) at particular time values. Time and rate information is given in the table below: t minutes r(t) liters/minute (a) (3pts) Plot the values of r(t) for each of the above time values in the axes below (b) (5pts) Using the values given in the table with subintervals of 5 minutes, compute an upper estimate for the area under the curve r(t). If it is helpful to you, you may draw rectangles in the plot from part (a). (c) (3pts) In the context of this problem, write one to two sentences describing what the value you found in part (b) represents. Be sure to include appropriate units in your explanation as needed. (d) (3pts) What does 30 0 r(t) dt appropriate units in your explanation as needed. represent in the context of this problem? Be sure to include

21 5. (8pts) The velocity of a particle moving back and forth along a number line is given by the equation: v(t) = 3 t 1 2 sin(t). (a) Determine the function that gives the position, s(t), of the particle at time t if s(0) = 0. Evaluate all trigonometric functions exactly. s(t) = (b) (4pts) Using the function you found in (a), determine the position of the particle when t = π 2. Evaluate all trigonometric functions exactly. If you express your final answer in decimal form, round to at least 4 decimal places.

22 6. (15pts) In the axes provided, sketch the graph of a function that has the properties listed in the table as well as the integral property. Function and Derivative Properties: f(x) f (x) f (x) x < 4 f (x) > 0 f (x) > 0 x = 4 lim x 4 4 < x < 2 f (x) < 0 f (x) > 0 2 < x < 0 f (x) > 0 f (x) > 0 x = < x < 3 f (x) > 0 f (x) < 0 x > 3 lim f(x) = 0 x Integral Property: 1 3 f(x) dx < 0

23 7. You have been provided with 100 feet of fencing to create an enclosed garden with maximal area. One of your neighbors, Douglas, suggested you split the fencing around two areas, one circular and one square. Your other neighbor, Katherine, insists that you only need one of those (though she doesn t specify which). Let s represent the side length of the square. Let r represent the radius of the circle. (a) (2pts) Write an equation that expresses the combined perimeter of both shapes in terms of s and r. Combined Perimeter Equation: (b) (2pts) Write an equation that expresses the combined total area of both shapes in terms of s and r. Combined Area Equation: (c) (2pts) Using interval notation, provide the domain of values for the side length of the square, s. (d) (2pts) Using interval notation, provide the domain of values for the radius of the circle, r. Keep values in exact form (no decimals). (e) (12pts) Use calculus to determine the values of s and r which result in maximal area for your garden. Be sure to demonstrate that the values you found result in a maximum area. If you use decimals, do not round until your final answers and round to at least 4 decimal places. [Hint: As it turns out, Katherine is correct, but calculus must be used to show this!]

24 8. (4pts) Sam evaluated the following integral. If Sam correctly evaluated the integral, draw a smiley face. If Sam did not correctly evaluate the integral, explain the error(s) that Sam made. cos(3x) dx = 1 3 sin(3x) + C 9. (4pts) Wes evaluated the following integral. If Wes correctly evaluated the integral, draw a smiley face. If Wes did not correctly evaluate the integral, explain the error(s) that Wes made. π 2 dx = π3 3 + C 10. (4pts) Hilary evaluated the following integral. If Hilary correctly evaluated the integral, draw a smiley face. If Hilary did not correctly evaluate the integral, explain the error(s) that Hilary made. v dv = 3 2 v3/2 + C

25 11. (15pts - 5pts each) Indicate whether each of the following statements is True or False. If the statement is true, explain how you know it s true. If it is false, give a counterexample and explain why it is a counterexample. (A counterexample is an example of a function for which the if part of the statement is true, but the then part is false.) A graph with an explanation can be used as a counterexample. (a) An antiderivative of a product of functions, fg, is an antiderivative of f times an antiderivative of g. (b) If p (x) = q (x), then p(x) = q(x). (c) If f(x) is increasing on [2, 3], then 3 2 f(x) dx > 0.

26 NAME: Instructor: Time your class meets: Math 160 Calculus for Physical Scientists I Final Exam Wednesday, December 17, 7:30am-9:30am How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? -Albert Einstein 1. Turn off your cell phone and other devices (except your calculator). 2. Write your name on every page of the exam. Write your instructor s name on the cover sheet. 3. You may use a calculator on this exam. You must provide your own calculator; you may not use a laptop computer or smart phone. 4. No notes or other references, including calculator manuals or notes stored in calculator memory, may be used during this exam. 5. Use the back of the facing pages for scratch work and for extra space for solutions. Indicate clearly when you wish to have work on a facing page read as part of a solution to a problem. HONOR PLEDGE I have not given, received, or used any unauthorized assistance on this exam. Furthermore, I agree that I will not share any information about the questions on this exam with any other student before graded exams are returned. (Signature) (Date) Please do not write in this space. 1. (18pts) 2-3. (12pts) 4. (12pts) 5. (13pts) 6. (8pts) 7. (12pts) 8. (13pts) 9. (12pts) TOTAL May the scwartz be with you

27 1. (18pts - 3pts each) Evaluate the following limits, derivatives, and integrals as instructed. L Hopitals Rule is not allowed. If an answer is or does not exist explain how you know. Answers will be graded as right or wrong. You will only receive credit if your answer is fully correct with supporting work. (a) lim x 3 x 3 x 2 + x 12 b 1 (b) lim b b (c) Find f (x) given f(x) = sin(x) tan(x 2 ) (d) Find g (t) given g(t) = 4 t 3t 2 + t 3 (e) ( 8x 4 4 ) x + 5 x + 1 dx 2 (f) π π/2 sin(x) (cos(x) + 2) 2 dx

28 Use the graph of f(x) below to answer the following questions. 2. (6pts) Order the following from smallest to largest (fill in the blanks below): 1 f(x) dx, 4 f(x) dx, f(x) dx < < 3. (6pts) The Riemann sum from using right-endpoints and 6 subintervals of equal length on [0, 6] is a way of computing an approximate value for (circle one) (a) 6 1 f(x) dx (b) 6 0 f(x) dx (c) 5 1 f(x) dx and the value will be (circle one) (a) Greater than zero. (b) Less than zero. (c) Equal to zero. (d) Cannot be determined.

29 4. (12pts) Let g(x) = x 0 f(t) dt where f is graphed below and g is defined for x 0: (a) Does g have any local maxima within (0, 6)? If so, where are they located? (Explain how you know.) (b) Does g have any local minima within (0, 6)? If so, where are they located? (Explain how you know.) (c) At what value of x does g attain its absolute maximum on (0, 6)? (Explain how you know.)

30 5. (13pts) Consider the curve defined by the equation x 2 + y 2 xy = 1 (a) Use implicit differentiation to find dy dx. (b) Verify that the points (1, 1) and ( 1 3, 1 ) are on the curve. 3 (c) Find the slope of the line tangent to the curve at each point. Slope at (1, 1) is Slope at ( 1 3, 1 ) is 3 The tangent lines are parallel / perpendicular / neither (circle one).

31 6. (8pts) Indicate whether each of the following statements is True or False. If the statement is true, explain how you know it s true. If it is false, give a counterexample and explain why it is a counterexample. (A counterexample is an example of a function for which the if part of the statement is true, but the then part is false.) A graph with an explanation in words can be used as a counterexample. (a) If a function, f(x), is continuous at x = 2, then it is also differentiable at x = 2. (b) Suppose f is a function such that f (x) < 0 for all x. Let g(x) = f(f(x)). Then g must be increasing for all x.

32 7. (12pts) Given two nonnegative real numbers whose sum is 9, find the maximum value of the product of one number with the square of the other number. Use calculus to justify your answer.

33 8. (13pts) In the axes provided, sketch the graph of a function f(x) that has the following properties. (a) lim f(x) = x (b) f (x) < 0 and f (x) < 0 for x < 4 (c) (d) lim f(x) = x 4 lim f(x) = x 4 + (e) f (x) < 0 and f (x) > 0 for 4 < x < 2 (f) f (x) > 0 for 2 < x < 0 (g) lim x 0 f(x) = (h) f (x) < 0 for 0 < x < 2 (i) f (x) > 0 for 0 < x < 3 (j) f (x) > 0 for 2 < x < 4 (k) f (x) < 0 for x > 3 (l) f (x) < 0 for x > 4

34 9. (12pts) The following is to be used for parts (a)-(c). Part (c) is on the next page. Below is the graph of the region bounded by the curves: f(x) = x 2 4x + 5, g(x) = 1 2, x = 1, x = 3 (a) Write out, but DO NOT evaluate the integral that will give the area of the region. (b) Write out, but DO NOT evaluate the integral that will give the volume of the solid generated by revolving the region about the x axis.

35 (c) A three-dimensional solid has a base that is the given region. The cross-sections of the solid are squares perpendicular to the x-axis (i.e. the length of a side of the square is the distance from f(x) to g(x)). See the graphs below for reference. Set up, but DO NOT evaluate the integral that gives the volume of the solid.