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Assignment 1 questions

# Assignment 1 questions - | x-1 | x-1 if x 6 = 1 2 if x = 1...

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STAT 0301 (B) Concise Calculus Sem 1 2007-08 ¥ ASSIGNMENT 1 (Due Oct 8, 2007) Q1. (2 Marks) Use the following revenue function and the cost function R ( x ) = x (86 - 5 x ) , C ( x ) = 156 + 19 x ( x > 0) to set up the profit function defined by P ( x ) = R ( x ) - C ( x ). For what values of x will : (a) a profit occur (i.e. P ( x ) > 0)? (b) a loss occur (i.e. P ( x ) < 0)? Q2. (2 Marks) Find the range of the function f ( x ) = ln(cos 2 x ) . Q3. (3 Marks) Solve the inequality: | 3 x - 1 | < | x + 2 | - 1. Q4. (3 Marks) Solve the equation: | (log 3 x ) - 4 | = 2 . Q5. (2 Marks) Below is the graph of the function f ( x ), find the following limits (if they exist) (a) lim x 1 + f ( x ) (b) lim x 1 + f ( x ) - f (2) - 2 [ f ( x )] 2 - 2 f ( x ) - 3 Q6. (3 Marks) Let f be the function defined by f ( x ) = ( 3 x + k, if x < 3 , 23 - 2 kx, if x > 3 , where k is a constant. How would you redefine the function f at x = 3 so that it is continuous at x = 3 ?

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Q7. (3 Marks) Using the fact that a polynomial is always continuous, solve the inequality ( x - 1)( x + 1) 2 ( x - 3)( x + 4) < 0 . Q8. (3 Marks) For the functions given in the following, find the limits, if they exist. (a) lim x 2 x - 2 x 3 - 8 (b) lim x →∞ x x - 1 - x (c) lim x 1 f ( x ) where f ( x ) =
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Unformatted text preview: | x-1 | x-1 , if x 6 = 1 , 2 , if x = 1 . 9. (3 Marks) Find the critical values of f ( x ) = x 4 / 3-x 1 / 3 . (a) Identify the intervals on which f is increasing. (b) Identify the intervals on which f is decreasing. (c) Are there any extrema? 10. (4 Marks) Using L’Hopital’s rule, ﬁnd the following limits, if they exist. (a) lim x → e x-(1 + x ) x 3 = (b) lim x →∞ x-4 / 3 sin(1 /x ) = (c) lim x → + 1-ln x e 1 /x = (d) lim x →∞ x + sin2 x x = 11. (2 Marks) A crystal has the shape of a perfect circular cone. Its height is 250 mm and is increasing at a rate of 4 mm per century. Its radius is 45 mm and is decreasing at a rate of 0.4 mm per century. (a) Is its volume increasing or decreasing? (b) At what rate? [Hint: You may use the fact that the volume of a circular cone is V = 1 3 πr 2 h , where h = height, r = radius.]...
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