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P MT 13 w sln

# P MT 13 w sln - f x = x | x | is diﬀerentiable and ﬁnd...

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Math 1A Midterm 1. 2004-9-30 2:00pm-3:30pm You are allowed 1 sheet of notes. Calculators are not allowed. Each question is worth 1 mark, which will be given only for a clear correct answer and correct working. There is no partial credit for wrong answers. There are questions on both sides of the paper. 1. Find the domain of the function f ( x ) = x - 4. 2. Sketch the graph of y = | x 2 - 1 | . 3. Find a formula for the inverse of the function y = e x . 4. Determine the infinite limit (+ or -∞ ) of lim x 0 x + 1 x 2 ( x - 1) . 5. Evaluate the limit lim x →- 1 x 2 - x - 2 x + 1 . 6. Find the constant c that makes g continuous for all reals, where g ( x ) = x 2 - c 2 if x < 4, g ( x ) = cx + 20 if x 4. 7. Find the numbers at which f is discontinuous, where f is defined by f ( x ) = x + 1 if x < 0, f ( x ) = e x if 0 x 1, f ( x ) = 2 - x if x > 1. 8. Evaluate lim x + 4 x 2 - 3 x 2 - x + 10 . 9. Find the equation of the tangent line to the curve y = x 3 at the point where x = 1. 10. Sketch the graph of a function for which f (0) = 0, f (0) = - 1, f (1) = 0, f (1) = - 1. 1

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11. Determine for what values of
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Unformatted text preview: f ( x ) = x | x | is diﬀerentiable and ﬁnd a formula for f . 12. Diﬀerentiate the function y = 6 x-8 / 3 . 13. Find all points on the curve y = x 3 + 3 x 2 + 3 x + 1 where the tangent is horizontal. 14. Diﬀerentiate ( x 3 + 1) e x . 15. Diﬀerentiate e x-1 e x + 1 . Answers: 1. x ≥ 4 2. Graph should come down to (-1 , 0), then go back up and come down again at (1 , 0), then go up again. At these 2 points the graph should have “corners”. 3. x = log( y ) 2 4.-∞ 5.-3 6. c =-2 7. x = 1 8. 4 9. y-1 = 3( x-1) or y = 3 x-2 10. Graph of function should pass though (0 , 0) and (1 , 0) and should slope down at both these points. 2 11. All x , derivative is 2 | x | . 12.-16 x-11 / 3 . 13. ( x, y ) = (-1 , 0). (Accept just x =-1 as correct.) 14. ( x 3 + 3 x 2 + 1) e x . 15. 2 e x / ( e x + 1) 2 3...
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