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Find the derivative of the following functions by using the difference quotient / first principle method. Determine the value of the derivative at...
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1. Find the derivative of the following functions by using the difference quotient / first principle method . Determine the value of the derivative at the specified value of x . a) 2 3 ) ( 2 x x x s , x = 8 b) x x f 1 ) ( , x = 10. (3+3 marks) 2 Find the equation of the tangent line to the curve x y at the point where 4 x . (4 marks) 3 A manufacturer estimates that when x units of a particular commodity are produced, the total cost will be x x C 55 50000 ) ( dollars, and furthermore that all x units will be sold at $420 per unit. a) Find the marginal cost and the marginal revenue. b) Estimate the cost of producing the ninth unit. c) Find the profit function. d) Estimate the marginal profit derived from the sale of the ninth unit. e) Find the actual profit derived from the sale of the ninth unit. (2+2+2+2+2 marks) 4. The population P ( t ) of a bacterial colony t hours after observation begins is found to be changing at the rate t t e e dt dP 03 . 0 1 . 0 150 200 . If the population was 200,000 bacteria when observations begin, what will the population be 12 hours later? (4 marks) 5. Find the area bounded by the graphs of f(x) = 3 2 1 x , g(x) = -x 2 + 1, x = -2, and x =1. (5 marks) 6. A manufacturer has found that marginal cost is 3q 2 – 60q + 400 dollars per unit when q units have been produced. (a) If the total cost of producing the first 2 units is $900. What is the total cost of producing the first 5 units?
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(b) Given that the fixed cost is $500, find the total cost function C(q) and the cost of producing 30 units? (3+3 marks)
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