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Quantitative Analysis

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Quantitative Business Analysis, Mathematics part, Autumn 2012
Exercise set 1. The due date for this warm-up exercise set is Monday 17.9. at 9:45 (before the first exercise group). Please submit your exercise answers to the course box on the 2nd floor of the Chydenia building. Problem 1. (a) Consider the graph of f (x) = x2 and the line tangent to f (x) at (a, f (a)). The equation of the 1 2 tangent line is y = − 5 x − 25 . Find a, f (a), and the slope of the parabola f (x) at (a, f (a)). 6 (b) Consider the graph of y = x3 . Find the point(s) on the graph where the slope is equal to 11 . (c) The demand function for a commodity with price P is given by the formula D(P ) = a − bP . Find dD(P )/dP . (d) The cost of producing x units of a commodity is given by the formula C(x) = p + qx2 . Find C (x), the marginal cost. Problem 2. (a) Determine the limit limx→0 (3 + 2x2 ). (b) Determine the limit limx→2 (2x2 + 5)3 2 (c) Determine the limit limx→1 x +7x−8 (tip: modify the numerator using similar approach as in the x−1 lecture example) (d) Let f (x) = 4x2 . Show that f (5 + h) − f (5) = 40h + 4h2 . Hence, f (5 + h) − f (5) = 40 + 4h h Using this result, find f (5). Problem 3. Use differentiation rules (not the formal definition) to do the following: (a) Differentiate y = √ 3 − 6x2 + 49x − 54 2x 2 (b) Differentiate y = x − x 2 (c) Differentiate y = (x + 3x − 5)3 (d) Find the equation of the tangent line to the graph of f (x) = x/(x2 + 2) at x0 = 3. Tip: Given a point on a line and the slope, one can construct an equation of the line. Problem 4. Give an example of a function (if possible), which is defined for all real numbers and (a) is not continuous and not differentiable at x = 0 (b) is continuous, but not differentiable at x = 0 (c) is differentiable, but not continuous at x = 0 Problem 5. (Optional bonus question) (a) Use the formal definition of derivative, i.e. f (x) = limh→0 f

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