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Unformatted text preview: Differential Calculus Winter 2009-2010 Final Test 1. Find the limit (if it exists). x2 - 5x + 6 a) lim x3 x2 - 9 x-1-1 b) lim x2 x-2 c) lim 2. a) Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? x2 + 3, x 1 f (x) = x - 5, 1 < x 2 -3, x>2 b) Find vertical asymptotes (if any) of the function. f (x) = 3. Find the defivative by the limit process f (x) = 3x2 + 10x - 2010 x2 x+1 - 2x - 3 1 - cos x x0 5x 4. Find the derivative of the function a) f (x) = b) c) f (x) = cos2 x x2 - 5x + 2 x2 + 1 f (x) = (sin x) x2 - 1 5. Find an equation of the tangent line to the graph of f at the indicated point f (x) = x2 + 5, (2, 3) 6. Find dy/dx by implicit differentiation and evaluate the derivative at the point (1, 0) x sin y + cos y = y + x 7. Determine whether Rolle's Theorem can be applied to f on the closed interval [-1, 3]. If the theorem can be applied, find all values of c in the open interval (-1, 3) such that f (c) = 0 f (x) = (x - 3)(x + 1)2 8. Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. f (x) = x3 + 6x2 - 15x + 1 9. Find the points of inflection and discuss concavity of the graph of the function. f (x) = x4 + 4x3 - 18x2 + 100x - 1 10. Find two positive integers that satisfy the given requirements: The sum of the first and twice the second is 20 and the product is a maximum. Bonus Problems 11. A right triangle is formed in the first quadrant by the x- and y- axes and a line through the point (1, 2). Find the vertices of the triangle such that its area is a minimum. 12. Let p(x) = x4 + ax2 + 1. Determine all values of a constant a such that p has exactly one relative maximum. ...
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- Spring '11
- Calculus, Rolle, lim x3 x2, lim x2 x-2