# Final Exam Solution Winter 2013 - University of Toronto...

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University of TorontoFACULTY OF APPLIED SCIENCE AND ENGINEERINGSolutions toFINAL EXAMINATION, DECEMBER, 2013Duration:2 and 1/2 hoursFirst Year - CHE, CIV, IND, LME, MEC, MMSMAT186H1F - CALCULUS IExam Type: AGeneral Comments:1. In Question 1(c) many students thoughtsin(πt)dt=-πcos(πt)+Cor-cos(πt)+C.2. In Question 2(b) your calculator should be in radian mode.If your calculator is indegree mode otherwise correct calculations will give youx= 0.999847741. . .which isnotthe correct solution to cosx=x.3. The graph in Question 6 isnota circle, although this was a very popular misconception.Question 6 can be greatly simplified using symmetry. Computationally it is easier tocalculate the derivatives implicitly, although explicit calculations are possible if youkeep in mind which quadrantxandyare in.4. Not surprisingly the optimization problem, Question 7, caused the most difficulty, asmany students could not set it up correctly.5. It was astonishing how many students threw marks away because of bad notation orby not using equal signs. Many students would profit greatly if they cleaned up theirpresentation!Breakdown of Results:434 students wrote the exam. The marks ranged from 20% to100%, and the average was 65.5%. Some statistics on grade distributions are in the table onthe left, and a histogram of the marks (by decade) is on the right.Grade%Decade%90-100%4.9%A19.4%80-89%14.5%B21.4%70-79%21.4%C26.3%60-69%26.3%D19.6%50-59%19.6%F13.3%40-49%8.5%30-39%3.2%20-29%1.6%10-19%0.0%0-9%0.0%
1. [15 marks] Find the following:(a) [5 marks]ex+11 +x2+1x+ sinhxdx
(b) [5 marks]π/40tan2xsec2x dx.