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MATH 110 Midterm: Math 110 Exam 2 V4

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turquoisegnu128

31 Jan 2019

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Mar 02, 2011: (30 points, find the fourth derivatives of (i) f (x) = 5x + 3 x2 + 1: y = ( x + 1)(x2 + 1, (30 points) a spherical balloon is increasing at a constant volume rate of 7 cubic inches per minute. It is determine that the monthly demand for its best selling toy increase by 4000 units for every sh. 10 reduction in price. Use this to nd the demand function and total revenue function. Find the marginal revenue per toy when x = 10000: (30 points) find the open intervals on which each the following functions is increas- ing/decreasing. Then nd all relative extremas of the function.

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1. (30 pts) (I) Calculate the derivatives of the following functions. NO NEED TO SIMLIFY ! (a) (8 pts) y = x tan-1 (x5) (b) (8 pts) y = xsin x (II) (14 pts) The curve with equation is a "rotated ellipse". x2 + xy + y2 = 4 (a) Use implicit differentiation to find the derivative (b) Consider the two points (3, -1) and (2, -2). Show that one of these points lies on the ellipse defined above, and one does not. (c) For the point in part (b) which lies on the ellipse, find the equation of the line tangent to the ellipse at this point.

2.6,#2(a), (b), 13(a), 17(a), 20(a), 25(a), 29, 30

Ed 9th B Blackboarc x h, ed rds.pd Adobe Read Wind 178 1334 100% Tools Sign (b) If 0 is increasing at the rate of radian per minute, find the when rates of change of the area when 0 /6 and 6 m/3 Explain why the rate of change of the area of the triangle is when not constant even though d6/dt is constant. 16. Volume The radius of a sphere is increasing at a rate of when 3 inches per minute a) Find the rates of c hange of the volume when 9 inches when and 36 inches b) Explain why the rate of change of the volume of the sphere when is not constant even though dr/dt is constant. 17. Volume A spherical balloon is inflated with gas at the rate of when 3, y 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is a) 30 centi and (b) 60 centimeters when 4, y 18. Volume All edges of a cube are expanding at a ate of 6 centimeters per second. How fast is e volume c hanging In Exercises 5-8, a point is moving along the graph of the given hen each edge is (a) 2 centimeters and (b) 10 centimeters? function such that dxldt is 2 centimeters per second. Find dyldt for the given values of x 19. Surface Area The conditions are the same as Exercise 18 Determine how fast the surfa changing when each edge is (a) 2 centimeters and (b) 10 centimeter 20. Volume The formula for the volume of a cone is V Find the rates of change of the volume if dr/dt is 2 inches per minute and h 3r when (a 6 inches and (b 7. y 21. Volume At a sand and gravel plant sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately WRITING ABOUT CONCEPTS three times the altitude. At what rate is the heigh of the pile changing when the pile is 15 feet high? 9. Consider th e li near function y b. If changes at a constant rate, does y change at a constant rate? If so, does 22. Depth A conical tank (w th vertex down s 10 feet across th change at the same rate as x? Explain. top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth 10. In your ow words, state the guidelines for solving related of the water when th e water is 8 feet rate problems 23. Depth A swimming pool is 12 meters long, 6 meters wide 1 meter deep at the shallow end, and 3 meters deep at the deep 11. Find the rate of change of the distance between the origin end (see figure on next page Water is being pumped nto th f dx/di and a moving point n the graph of y pool at L cubic meter per minute. and there is 1 meter of water centimeters per second the deep end. 12. Find the rate of change of the distance between the origin (a) What percent of the pool is filled? f dx/dt and a moving point on the graph of y b) At what ate is th e water leve sing? centimeters per second 13. Area The radius of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area. when (a 8 centimeters and (b 32 centimeters Comment a A Lueth 18 Research Help 6:30 PM 6/4/2013

Show transcribed image text Ed 9th B Blackboarc x h, ed rds.pd Adobe Read Wind 178 1334 100% Tools Sign (b) If 0 is increasing at the rate of radian per minute, find the when rates of change of the area when 0 /6 and 6 m/3 Explain why the rate of change of the area of the triangle is when not constant even though d6/dt is constant. 16. Volume The radius of a sphere is increasing at a rate of when 3 inches per minute a) Find the rates of c hange of the volume when 9 inches when and 36 inches b) Explain why the rate of change of the volume of the sphere when is not constant even though dr/dt is constant. 17. Volume A spherical balloon is inflated with gas at the rate of when 3, y 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is a) 30 centi and (b) 60 centimeters when 4, y 18. Volume All edges of a cube are expanding at a ate of 6 centimeters per second. How fast is e volume c hanging In Exercises 5-8, a point is moving along the graph of the given hen each edge is (a) 2 centimeters and (b) 10 centimeters? function such that dxldt is 2 centimeters per second. Find dyldt for the given values of x 19. Surface Area The conditions are the same as Exercise 18 Determine how fast the surfa changing when each edge is (a) 2 centimeters and (b) 10 centimeter 20. Volume The formula for the volume of a cone is V Find the rates of change of the volume if dr/dt is 2 inches per minute and h 3r when (a 6 inches and (b 7. y 21. Volume At a sand and gravel plant sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately WRITING ABOUT CONCEPTS three times the altitude. At what rate is the heigh of the pile changing when the pile is 15 feet high? 9. Consider th e li near function y b. If changes at a constant rate, does y change at a constant rate? If so, does 22. Depth A conical tank (w th vertex down s 10 feet across th change at the same rate as x? Explain. top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth 10. In your ow words, state the guidelines for solving related of the water when th e water is 8 feet rate problems 23. Depth A swimming pool is 12 meters long, 6 meters wide 1 meter deep at the shallow end, and 3 meters deep at the deep 11. Find the rate of change of the distance between the origin end (see figure on next page Water is being pumped nto th f dx/di and a moving point n the graph of y pool at L cubic meter per minute. and there is 1 meter of water centimeters per second the deep end. 12. Find the rate of change of the distance between the origin (a) What percent of the pool is filled? f dx/dt and a moving point on the graph of y b) At what ate is th e water leve sing? centimeters per second 13. Area The radius of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area. when (a 8 centimeters and (b 32 centimeters Comment a A Lueth 18 Research Help 6:30 PM 6/4/2013

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INSTRUCTIONS: There are 15 problems on this exam, and each problem is worth 13 points, plus 5 points for writing down an approximate volume of the planet Earth in cubic miles at the bottomleft corner of page 3, for a total possible 200 points. For this exam, you are allowed to use one 8.5"X11" sheet of notes (both sides, in your own handwriting), pencil, eraser, ruler, scratch paper and calculator. Even though the questions are multiple-choice, you need to show enough work to justify your answer in order to receive full credit. After you have determined the correct solution to each problem, write the letter corresponding to your answer in the appropriate space on the right side of the page AND circle your answer. If your answer does not agree with any of the choices, after double-checking your work, choose option E) and prominently mark your answer (box, circle, underline, etc.). Time limit: Two hours. Attach (i.e. staple) any notecard and scratch paper to the rear of this exam paper. Good luck! MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. The northern third of Indiana is a rectangle measuring 96 miles by 132 miles. Thus, let D = [0, 96] times [0,132]. Assuming that the total annual snowfall (in inches), S(x,y), at (x,y) euro D is given by the function S(x,y) = 60e-0.001 (2x + y) with (x,y) euro D, find the average snowfall on D. 51.14 inches 52.06 inches 51.78 inches 52.44 inches Evaluate the work done between point 1 and point 2 for the conservative field F. Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. Evaluate the surface integral of G over the surface S. S is the portion of the cone Find the flux of the vector field F across the surface S in the indicated direction. Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. Integrate the function f over the given region. f(x,y) = 1/ln x over the region bounded by the x-axis, line x = 9, and curvey y = ln x Reverse the order of integration and then evaluate the integral Use a double integral in polar coordinates to find the area of the region specified in polar coordinates. the region inside both r = 7 sin Theta and r = 7 cos Theta Use spherical coordinates to find the volume of the indicated region. the region bounded above by the sphere x2 + y2 + z2 = 25 and below by the cone z = Find the center of mass of the rectangular solid of density (x,y,z) = xyz defined by 0 LE x LE 8, Use the given transformation to evaluate the integral. where R is the parallelogram bounded by the lines y = 8x + 8, y = 8x +9, y = -6x + 2, y = -6x +7 Evaluate the line integral along the curve C. Find the work done by F over the curve in the direction of increasing t.