MTH 132 Solutions to Exam 2 Nov. 23rd 2015

Transcription

1 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 9. 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 9 minutes for this exam. I have read and understand the above instructions:. SIGNATURE Page 1 of 9

2 Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 1. Consider f(x) = x + x 4. (a) ( points) Find the vertical asymptote(s) of f(x) or explain why none exist. Solution: x = 1 (b) (6 points) Find the slant asymptote(s) of f(x) or explain why none exist. Solution: x + x 4 So the slant asymptote is y = x +. = (x x + 1) + (4x 4) 1 = () + 4() 1 = (1 points) Find the absolute minimum and maximum of f(x) = x 9x + 11 on the interval [ 1, ]. Solution: f (x) = 6x 18x = 6x(x ) so f has critical points at x =,. Since is not in the interval we need only consider f( 1) = = f() = 11 f() = = 9 So 11 is the absolute max and 9 is the absolute min. Page of 9

3 . (a) (4 points) For a differentiable function f(x), present the formula of its linearization at x = a. Solution: L(x) = f(a) + f (a)(x a) (b) (1 points) Use linearization to estimate 8.1 Solution: Take f(x) = x and a = 8 then f(8) = f (x) = 1 x / f (8) = 1 (4) = 1 1 Assembling it all together we get L(x) = + 1 (x 8) so we get the final answer of: 1 L(8.1) = + 1 (8.1 8) 1 = (.1) = = 41 1 Page of 9

4 4. (a) (8 points) Let f(x) = Solution: x 1 + t dt. Find f (x). f (x) = 1 + (x ) (x) = x 1 + x 4 (b) (8 points) Evaluate Solution: 1 (x 6x 4 + 5) dx. 1 ] 1 [ (x 6x 4 + 5)dx = x 6 5 x5 + 5x = [1 65 ] + 5 [] = = (1 points) Find (a) the upper sum and (b) the lower sum, when we estimate the area under the graph of f(x) = 1 + sin x from x = to x = π using four rectangles of equal width. Solution: x = (π )/4 = π/. x π/ π π/ π f(x) (a) The upper sum is π ( ) = π. (b) The lower sum is π ( ) = π. Page 4 of 9

5 6. ( points) A total of 1 cm of material is to be used to make a box with no top. Assume that the base is to be twice as long as it is wide. Find the largest volume of such a box. Solution: With the above information we can tell that: 1 = lw + lh + wh l = w 1 = w + 4wh + wh 1 w = 6wh 1 w = h 6w And we want to maximize volume V = lwh. Using the above info we get V = lwh ( ) 1 w V (w) = (w)w 6w 1w w V (w) = = 4w w with w (, ) so then V (w) = 4 w giving us the only critical number w = 1 (within our domain). Now consider the number line (+) ( ) V 1 So therefore the maximum volume occurs at w = 1 and it is: V (1 ) = 4(1 ) (1 ) = 4 4 = 8 cm Page 5 of 9

6 7. Suppose f(x) = x 1, f (x) = x, f (x) = (x 6) x x 4 x 5 (a) (4 points) Write down the vertical asymptote and the horizontal asymptote of the curve y = f(x) (no supporting discussion is needed). Solution: V.A.: x = H.A.: y = (b) (1 points) Identify the intervals over which f(x) is increasing and decreasing, concave up and concave down. Write your answer in interval notation in the boxes below Solution: f has crit. points at x =,,. Consider the number line: ( ) (+) (+) ( ) f Therefore f is increasing on (, ) (, ) and decreasing on (, ) (, ). To check concavity consider the points at x = 6,, 6 and the number line: ( ) (+) ( ) (+) f 6 6 Therefore f is concave up on ( 6, ) ( 6, ) and concave down on (, 6) (, 6). (c) (6 points) Identify all values of x where f(x) attains its local maximum or minimum. Solution: f attains a max at x = and a min at x = (d) (4 points) Identify all values of x that are inflection points. Solution: f has inflection points at x = 6 and x = 6 (e) (4 points) By using the information in parts (a)-(d), sketch the curve of y = f(x). Solution: Page 6 of 9

7 Fill in the Blanks. No work needed. No partial credit available. 8. (4 points) If x = c is a critical number of f(x) = tan x sec x, and sec c, then tan c =1/. 9. (4 points) 6 x dx = x x 1. (4 points) lim x 4x x + = 1/ (4 points) If 4 1 f(x) dx = 5, and 4 f(x) dx =, then 1 (f(x) ) dx =1. Extra Work Space. Page 7 of 9

8 Multiple Choice. Circle the best answer. No work needed. No partial credit available. sin x 1. (6 points) Evaluate lim x x A. 1 B. 1 C. D. E. 1. (6 points) If we use Newton s method to approximate the root of f(x) = x 4x + 4, starting with x 1 =, then A. x = 1.5 B. x = 1.6 C. x = 1.7 D. x = 1.8 E. x = (6 points) Evaluate A. n n B. n + n C. n + n D. n + n E. n + 4n n (k + 1) k=1 15. (6 points) Suppose f(x) = x x on the interval [1, ]. Which of the following is guaranteed by the Mean Value Theorem? A. There is c in (1, ) such that f (c) =. B. There is c in (1, ) such that f(c) =. C. There is c in (1, ) such that f (c) =. D. There is c in (1, ) such that f(c) =. E. The Mean Value Theorem cannot be applied to f(x). Page 8 of 9

9 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: 15 Page 9 of 9