Name: Math 1120, Final. December 12, Net id: PLACE AN X IN THE BOX TO INDICATE YOUR SECTION

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1 Math 1120, Final December 12, 2017 Name: Net id: PLACE AN X IN THE BOX TO INDICATE YOUR SECTION Ian Lizarraga MWF 8:00 8:50 Ian Lizarraga MWF 9:05 9:55 Swee Hong Chan MWF 12:20 1:10 Teddy Einstein MWF 2:30-3:20 My Huynh MWF 3:35-4:25P Shihao Xiong TR 8:40-9:55 Marcelo Aguiar TR 1:25 2:40 Marcelo Aguiar TR 2:55 4:10 INSTRUCTIONS Enter your name, net id and mark your section now. This test consists of 12 pages (besides this cover sheet). Look over this test as soon as the exam begins. If you find any missing pages, please ask a proctor for another copy. Show your work. To receive full credit, your answers must be neatly written, and logically organized. Explain all steps that are not self explanatory to an average student. If you need more space, write on the back side of the preceding sheet, but be sure to label your work clearly. Scrap paper is available for rough work. You may not hand in work on scrap paper. You have 150 minutes to complete this exam. This is a closed book exam and no notes are allowed. You are not allowed to use a calculator, cell phone, or any other electronic devices. Academic integrity is expected of all Cornell University students at all times, whether in the presence or absence of members of the faculty. Understanding this, I declare I shall not give, use, or receive unauthorized aid in this examination. Please sign below to indicate that you have read and agree to these instructions. OFFICIAL USE ONLY (do not fill in) 1. / / / / / / / / / / / / 10 Total: / 156 Signature of Student

2 Math 1120 (Fall 2017) Final Exam (12/12/2017) 1 1. Calculate the following integrals: (a) x 3 ln x dx (b) (t + 3)e t2 +6t+5 dt (c) x 3 9 x 2 dx

3 Math 1120 (Fall 2017) Final Exam (12/12/2017) 2 2. Let R be the region bounded by the line y = x + 3 and the parabola y = x 2 x. Express the following quantities in terms of integrals. In this exercise, you are not asked to compute any integrals, only to set them up. (a) The area of R. (b) The volume of the solid formed by revolving R about the line x = 4. (c) The volume of the solid whose base is R and whose cross sections perpendicular to the x axis are squares (with the side of the square going from the parabola to the line).

4 Math 1120 (Fall 2017) Final Exam (12/12/2017) 3 3. Consider the function f(x) = (a) Compute f (x). x 1 t 2 e t2 1 dt. (b) Find the length of the curve y = f(x) for 1 x 2. (c) Express lim f(x) as an integral. (You do not need to compute its value.) x +

5 Math 1120 (Fall 2017) Final Exam (12/12/2017) 4 4. Consider the differential equation dy dx = y3 2y 2 + y. (a) Build the phase line, determine the stable and unstable equilibrium values, and sketch some solution curves. (b) Assume y(0) = 1/3. Find lim x y(x).

6 Math 1120 (Fall 2017) Final Exam (12/12/2017) 5 5. Solve the differential equation with initial condition dy dx = x2 (y 2 + 1), y(1) = 3.

7 Math 1120 (Fall 2017) Final Exam (12/12/2017) 6 6. (a) Carefully calculate the improper integral x 1 dx. (b) Determine whether the improper integral 1 sin 2 x x 2 dx converges. (Explain.)

8 Math 1120 (Fall 2017) Final Exam (12/12/2017) 7 7. Determine if the following series converge or not, and if they converge absolutely or not. Provide a careful explanation in each case. (a) ( 1) n n n n=1 (b) ( 1) n sin(n3 ) n 2. n=1

9 Math 1120 (Fall 2017) Final Exam (12/12/2017) 8 8. (a) Find the first 3 nonzero terms in the power series representation of sin(x 2 ). (b) Calculate lim x 0 sin(x 2 ) x x6 x 6. (c) Use (a) to find the first 3 nonzero terms in the power series representation of x cos(x 2 ).

10 Math 1120 (Fall 2017) Final Exam (12/12/2017) 9 9. Find the radius and interval of convergence of the following series. (a) n=0 ( 1) n 4 n xn. (b) n=0 n + 1 (2n + 1)4 n xn.

11 Math 1120 (Fall 2017) Final Exam (12/12/2017) Consider the series n=3 1 n(n + 1). (a) Use a comparison test to determine if the series converges or diverges. (Explain.) (b) Find the sum of the series.

12 Math 1120 (Fall 2017) Final Exam (12/12/2017) In this exercise, explanations are not needed. ( 1) n Consider the series 2n + 1. n=0 0 1 (a) Locate in the picture the partial sums S 0, S 1, S 2, S 3, S 4 and S 5. (b) Locate approximately in the picture the sum of the series S. (c) Indicate whether each of the following 3 sequences has the stated property by filling in the table below with Y (yes) or N (no). sequence increasing decreasing neither increasing nor decreasing convergent S 0, S 2, S 4,... S 1, S 3, S 5,... S 0, S 1, S 2, S 3,...

13 Math 1120 (Fall 2017) Final Exam (12/12/2017) Consider the harmonic series n=1 1. Identify whether each of the following arguments is n valid or not, and explain why. (The question is whether the reasoning is valid, not whether the conclusion is true.) (a) We apply the ratio test. We have By the ratio test, the series converges. a n+1 1/(n + 1) = = n a n 1/n n + 1 < 1. (b) We apply the ratio test. We have By the ratio test, the series diverges. a n+1 1/(n + 1) = = n 1 as n. a n 1/n n + 1

Math 113 Winter 2005 Departmental Final Exam

Math 113 Winter 2005 Departmental Final Exam Name Student Number Section Number Instructor Math Winter 2005 Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems 2 through are multiple