Find the points on the ellipse 4x2 y2 4 that are farthest

Question

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 lim_x→∞ 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

lim_t→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?