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[Calculus 1321] Final Practice Exam

[Calculus 1321] Final Practice Exam - MATH 1321 Calculus I...

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MATH 1321 – Calculus I Sections 10 and 14 Fall 2004 Final Exam – Preparation 1. Find the following limits: (a) lim x 2 - x 2 - x - 2 | x - 2 | (b) lim x 7 x + 2 - 3 x - 7 2. Use the Squeeze Theorem to find lim x 0 e x 2 sin(1 /x ) . 3. Find the following limits: (a) lim x →∞ 1 - x - 3 x 2 2 x 2 - 7 (b) lim x →∞ 9 x 6 - x x 3 + 1 (c) lim x →∞ ( x 2 + x - x ) 4. Which of the following functions f has a removable discontinuity at a ? If the discon- tinuity is removable find an everywhere continuous function F such that f ( x ) = F ( x ) for all x = a . (a) f ( x ) = x 2 - 2 x - 8 x + 2 , a = - 2 (b) f ( x ) = x - 7 | x - 7 | , a = 7 (c) f ( x ) = x 3 + 64 x + 4 , a = - 4 5. Find the vertical and horizontal asymptotes of the curve y = x 2 - 9 2 x - 6 , x > 3. 6. Use the definition of the derivative to find f ( x ) if f ( x ) = 1 + x . 7. Use the definition of the derivative to find f ( x ) if f ( x ) = x x + 1 . 8. Let f ( x ) = sin 2 x . Use the definition of the derivative to find f (0). 9. Let f ( x ) = x 2 if x 2 mx + b if x > 2 (a) Find the values of m and b that make f continuous everywhere. (b) Find the values of m and b that make f differentiable everywhere. 10. Use the definition of differentiability to show that f ( x ) = | x - 6 | is not differentiable at a = 6. 11. If f ( x ) = e x g ( x ), where g (0) = 2 and g (0) = 5, find f (0). 12. Let f ( x ) = h ( x ) x , x = 0. If h (2) = 4 and h (2) = - 3, find f (2).
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13. Let f ( x ) = g ( h ( x )). If h (3) = 6, h (3) = 4, g (3) = 2 and g (6) = 7, find f (3). 14. Find the derivative y : (a) y = 1 2 π e - x 2 / 2 (b) y = sin(3 x ) x (c) y = (tan x )(2 sin x ) 15. If y = ln( x 2 + y 2 ), find y . 16. Find equations of the tangent lines to the curve y = x/ ( x + 1) that are parallel to the line x - 2 y = 2. 17. Find an equation of the tangent line to the curve y = e x that passes through the origin. 18. Find the point on the parabola y = x 2 at which the slope of the tangent line is equal to the slope of the secant line between A ( a, a 2 ) and B ( b, b 2 ). 19. Find the slope of the tangent line to the curve x 3 y 2 - 2 y = e y - 1 at the point (1 , 0). 20. Let f ( x ) = x 2 / 3 . (a) Use the definition of the derivative to show that f (0) does not exist. (b) Find f ( x ) for x = 0. (Here you don’t have to use the definition of the derivative.
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