Mat135_PT_T1 - www.prep101.com 1 Mat135 Practice Test #1...

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Unformatted text preview: www.prep101.com 1 Mat135 Practice Test #1 Part A—Multiple-choice questions (50 marks/5 marks each). Quick Tip 1 Be sure to simulate test-taking conditions when you work through this practice test to reduce your chances of brain-freeze when you write the test. Don’t look at the solutions until after you’ve attempted every question. Give yourself a time limit of 90 minutes to complete this practice test. 1. Find lim x→2 a) b) c) d) e) 2. x 3 − x 2 − 8 x + 12 x 3 − 12 x + 16 . 8 6 10 5 6 9 6 10 If y = 102x + 1, then dy = dx a) (2x + 1)102x+ 1 b) (2ln 10)102x + 1 ⎛ 2 ⎞ 2x + 1 c) ⎜ ⎟ 10 ⎝ ln 10 ⎠ d) 102x + 1 e) (102x + 1)(2) 3. Find a formula for the inverse, f −1(x), of the function given by f(x) = ln(x − 2). a) b) c) d) e) ln(x + 2) ln(−x + 2) −ln(x − 2) 2 + ex 2 − ex Solutions: Go to www.prep101.com, select University of Toronto, click on Free Stuff under Spotlight on right menu. www.prep101.com 4. Which of the following has a graph that is the graph of f(x) = 2e 3 x − 4 shifted to the left by 4 units and then shifted upward 5 units? a) b) c) d) e) 5. 2 Evaluate the lim e −1 / x . x →0 + a) b) c) d) e) 6. y = 2e 3 x +12 + 1 y = 2e 3 x − 4 + 1 y = 2e 3 x −12 + 1 y = 2e 3 x + 4 + 1 y = 2e 3 x −12 − 9 0 ∞ −∞ 1 Does not exist Determine lim x →∞ x 4x + 2 . a) ∞ 1 b) 2 1 c) 4 d) 1 e) Does not exist Solutions: Go to www.prep101.com, select University of Toronto, click on Free Stuff under Spotlight on right menu. 2 7. www.prep101.com At what point on the curve y = x + ln x is the tangent line parallel to the line y = 3x − 5? a) at x = 3 1 b) at x = 2 1 c) at x = 3 1 d) at x = 4 1 e) at x = 15 8. Determine the x-coordinates of the point on the curve y = 7x where the slope of the tangent line is equal to ln 49. a) 1 b) 2 c) 7 ln 2 d) ln 7 e) ln 14 9. Find y’if y = xcosx. ⎛ cos x ⎞ a) x cos x ⎜ − sin x ln x ⎟ ⎝x ⎠ cos x b) − sin x ln x x cos x c) x cos x d) x (sin x ln x ) ⎛ cos x ⎞ e) x cos x ⎜ ⎟ ⎝x⎠ Solutions: Go to www.prep101.com, select University of Toronto, click on Free Stuff under Spotlight on right menu. 3 10. 4 www.prep101.com Find the line passing through the point (−3, 0) and tangent to the curve y = x − 1 at some points. 1 (x + 3). 3 1 y = (x + 3). 5 1 y = ( x − 3). 4 y = 4(x − 3). 1 y = (x + 3). 4 a) y = b) c) d) e) Quick Tip 2 When writing the test, solve the easy questions first. This will build your confidence and ensure that you don’t run out of time with the easy questions unanswered. Part B—Short answer questions (50 marks) Quick Tip 3 When answering long and short answer questions (i.e. any question that isn’t multiple choice), don’t erase anything. Even if you don’t know how to complete the question, write something down (an attempt, a strategy, a guess) and give your professor a reason to give you part marks. 11. (6 marks) Let f(x) = 3x3. Find f ′(x) from the first principles (i.e by using only the definition of the derivative). Solutions: Go to www.prep101.com, select University of Toronto, click on Free Stuff under Spotlight on right menu. www.prep101.com 12. dy for each of the following. There is no need to dx simplify your final answers for this question. (16 marks) Use any suitable method to find (a) y = x arcsin x (b) f(x) = x2 e x (c) f(x) = 2 x3 ln x (d) f(x) = tan (x2 + π + x) Solutions: Go to www.prep101.com, select University of Toronto, click on Free Stuff under Spotlight on right menu. 5 www.prep101.com 13. 14. 6 ⎧2 x + c 2 if x < 3 Find the value of the constant c so that f is (6 marks) Let f ( x) = ⎨ . ⎩cx + c + 2 if x ≥ 3 continuous everywhere. (7 marks) For every number a, find an equation of the line which passes through the point (a, 0) and is tangent to the graph of y = ex. Solutions: Go to www.prep101.com, select University of Toronto, click on Free Stuff under Spotlight on right menu. 7 www.prep101.com 15. ⎛π ⎞ (7 marks) For the function y defined implicitly by sin(x − y) = y + 1, find y” at the point ⎜ ,0 ⎟ . ⎝2 ⎠ 16. (8 marks) Without using L′Hopital’s Rule, find the value of lim x →3 7 − 10 − x . 3− x Solutions: Go to www.prep101.com, select University of Toronto, click on Free Stuff under Spotlight on right menu. ...
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This note was uploaded on 03/15/2010 for the course MAT MAT135 taught by Professor Treung during the Fall '08 term at University of Toronto- Toronto.

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