Exam Solution Spring 2010 - University of Toronto Solutions to MAT 187H1S TERM TEST of Thursday February 4 2010 Duration 90 minutes General Comments 1

# Exam Solution Spring 2010 - University of Toronto Solutions...

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University of Toronto Solutions to MAT 187H1S TERM TEST of Thursday, February 4, 2010 Duration: 90 minutes General Comments: 1. In Questions 2 and 7 it really helps to make a suitable substitution before you use integration by parts. 2. Questions 2 and 3 are based on WileyPlus assigned homework problems. 3. Indefinite integral require a constant of integration; definite integrals do not. 4. Questions 1 and 3 require you to make a substituion into a definite integral. The correct way to do this is to change the limits of integration as you change the variable. Theorem 5.9.1 on page 391 of the textbook spells it out: b a f ( g ( x )) g ( x ) dx = g ( b ) g ( a ) f ( u ) du, where the substitution is u = g ( x ) . Doing this will actually save you work. 5. In Question 5, 6 x - x 2 0 ⇒ - 3 x - 3 3 , so letting x - 3 = 3 sec θ makes no sense at all. 6. In both Question 6(b) and 7 the substitution x = sec θ could be used to first simplify the integrand. 7. Any student who let dv = e x 4 dx, in Question 2, and then said v = e x 4 4 x 3 , should think of retiring from calculus. Breakdown of Results: 466 students wrote this test. The marks ranged from 16.7% to 100%, and the average was 67.5%. Some statistics on grade distributions are in the table on the left, and a histogram of the marks (by decade) is on the right. Grade % Decade % 90-100% 12.0% A 28.1% 80-89% 16.1% B 20.4% 70-79% 20.4% C 20.6% 60-69% 20.6% D 15.2% 50-59% 15.2% F 15.7% 40-49% 8.2% 30-39% 6.0% 20-29% 1.3% 10-19% 0.2% 0-9% 0.0%