Reference no: EM131076461
More Function Graphing; Optimization
Exercise 1- Let n be an arbitrary positive integer. Give an example of a function with exactly n vertical asymptotes. Give an example of a function with infinitely many vertical asymptotes.
Exercise 2- Let f be a function which is differentiable everywhere. Suppose that f'(x) > 1 for all x. Show that limx→∞ f(x) = ∞.
Exercise 3- Graph the function f(x) = x3 + 6x2 + 9x.
Indicate domain, critical points, inflection points, regions where the graph is increasing/decreasing, x-intercepts and y-intercepts, regions of concavity (up or down), local maxima and minima, any asymptotes and behavior at infinity.
Exercise 4- Find
limt→16(√t - 4/t - 16)
in three ways: (i) using methods learned up to and including the first midterm; (ii) by realizing the limit as f'(c) for some function f(t) and some value c; (iii) using L'Hospital's Rule.
Exercise 5- Find the point on the line y = 2x + 3 that is closest to the origin.
Exercise 6- Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).
Exercise 7- Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.
Exercise 8- At which points on the curve y = 1 + 40x3 - 3x5 does the tangent line have the largest slope?
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