MTH 132 Solutions to Exam 2 Apr. 13th 2015
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1 MTH 13 Solutions to Exam Apr. 13th 015 Name: Section: Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers. If you need scratch paper, use the back of the previous page. Without fully opening the exam, check that you have pages 1 through 8. Fill in your name, etc. on this first page. Show all your work. Write your answers clearly! Include enough steps for the grader to be able to follow your work. Don t skip limits or equal signs, etc. Include words to clarify your reasoning. Do first all of the problems you know how to do immediately. Do not spend too much time on any particular problem. Return to difficult problems later. If you have any questions please raise your hand and a proctor will come to you. There is no talking allowed during the exam. You will be given exactly 90 minutes for this exam. I have read and understand the above instructions:. SIGNATURE Page 1 of 8
2 MTH 13 Solutions to Exam Apr. 13th 015 Multiple Choice. Circle the best answer. No work needed. No partial credit available. 1. (6 points) Evaluate lim x A. 3 B. 3 C. 3 D. 3 E. Undefined 3x x + 5 x + 1. (6 points) The critical numbers of the function f(x) = 3 x (x 1) are: A. x = 0, 1/ B. x = 1/ C. x = 0 D. x = 0, 1/5 E. x = 1/5 3. (6 points) If we use Newton s method to approximate a root of f(x) = x 3 x 5, starting with x 1 =, then A. x =.9 B. x =.7 C. x =.5 D. x =.3 E. x = (6 points) Evaluate ( 1) k k A. 100 B. 50 C. 100 D. 50 E k=1 Page of 8
3 MTH 13 Solutions to Exam Apr. 13th 015 Fill in the Blanks. No work needed. No partial credit available. 5. ( points) The absolute maximum value of f(x) = x(3 x) on [0, 1] is, and it occurs at x =1. 6. ( points) If 5 f(x) dx = 7, and 5 f(x) dx = 3, then f(x) dx =. 7. ( points) 0 16 x dx =. (Hint: Use geometry.) 8. ( points) If f (x) = 8 3x and f(0) = 7, then f(1) =1. Extra Work Space. Page 3 of 8
4 MTH 13 Solutions to Exam Apr. 13th 015 Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 9. (a) (10 points) Find the linearization of f(x) = sin x at x = /6. We need the components Now we combine to get: f(/6) = 1/ f (x) = cos x f (/6) = 3/ L(x) = (b) (6 points) Use the result in (a) to estimate sin L(7/30) = ( x ) 6 ( ) 7. (Leave your answer in terms of ). 30 = ( 7 30 ) 6 ( 15) 10. (1 points) A particle is moving in a line with the acceleration a(t) = 3t + sin t. Given that v(0) = 1, find the velocity function v(t). v(t) = t 3 cos t + C v(0) = 1 + C = 1 C = v(t) = t 3 cos t + Page of 8
5 MTH 13 Solutions to Exam Apr. 13th (a) (6 points) State the mean value theorem. If f is continuous on [a, b] and differentiable on (a, b) then there exists a c (a, b) so that f f(b) f(a) (c) =. b a (b) (10 points) Use the mean value theorem to show that for any number a < b, it holds that sin b sin a b a f(x) = sin x is everywhere continuous and differentiable. Take arbitrary a < b then by the MVT there is a c such that f sin b sin a (c) = b a sin b sin a cos(c) = b a (Derivative of sin x) sin b sin a cos(c) = b a (Abs value of each side) sin b sin a 1 b a (cos x is bounded) b a sin b sin a (Alg.) 1. (1 points) Find (a) the upper sum and (b) the lower sum, when we estimate the area under the graph of f(x) = sin x from x = 0 to x = using four rectangles of equal width. Consider the sketches of sin x on the interval [0, ] below. Maxes Mins So we get: ( Upper Sum = ) = ( ) + ( Lower Sum = ) = ( ) Page 5 of 8
6 MTH 13 Solutions to Exam Apr. 13th (0 points) If a total of 300 cm of material is to be used to make a box with a square base and an open top, find the largest volume of such a box. Consider the picture to the right. We are trying to maximize Volume = V = x y The restricting equation is surface area: y Surface Area = 300 xy + x = 300 y = Giving a volume equation of one variable: To maximize lets find critical points: 300 x x V (x) = 1 ( 300x x 3 ) with x (0, 10 3) x x V (x) = 1 (300 3x ) (Never undefined so set to 0.) 0 1 (300 3x ) x = 100 x = 10 (Given domain.) We verify that x = 10 is a max since: Giving the largest volume of V (x) (+) ( ) ( ) V (10) = 1 (300(10) (10)3 ) = 1 (000) = 500 cm3 Page 6 of 8
7 MTH 13 Solutions to Exam Apr. 13th Suppose f(x) = x x 1, f (x) = 1 1 (x 1), f (x) = (x 1) 3 (a) (6 points) Write down the vertical asymptote and the slant asymptote of the curve y = f(x) (no supporting discussion is needed). V.A.: x = 1 S.A.: y = x + 1 (b) (10 points) Identify the intervals over which f(x) is increasing and decreasing, concave up and concave down. Rewriting f (x) we get: f (x) = (x 1) 1 (x 1) f (x) = x(x ) (x 1) f has critical points at 0, and is undefined at 1 so we check test points to get the picture: (+) ( ) ( ) (+) f 0 1 So f is increasing on (, 0) (, ) and decreasing on (0, 1) (1, ). Observing the second derivative it is undefined at x = 1. Choosing test points we get: ( ) (+) f So f is concave up on (1, ) and concave down on (, 1). 1 (c) (6 points) Identify all points (x, y) where f(x) attains its local maximum or minimum. By our work in (b) we know f has a local max at 0. Plugging this into our function we get the point (0, 0). f has a local min at. Plugging this into our function we get the point (, ). (d) ( points) Identify all values of x that are inflection points. By our work in (b) we would say there is an inflection point at 1. However since f is undefined there it is not an inflection point. There are no inflection points on f. 3 pts for answer x = 1. (e) (8 points) By using the information in parts (a)-(d), sketch the curve of y = f(x) Page 7 of 8
8 MTH 13 Solutions to Exam Apr. 13th 015 Congratulations you are now done with the exam! Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED. When you are completely happy with your work please bring your exam to the front to be handed in. Please have your MSU student ID ready so that is can be checked. DO NOT WRITE BELOW THIS LINE. Page Points Score Total: 150 Page 8 of 8
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