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Extra Problems for Chapter 7
11 An airplane travels in a straight line at 600 mph at an altitude of 4 miles. Find the rate of change of the angle of elevation one minute after the airplane passes directly over an observer on the ground. 12 A 40 ft ladder is to be propped up against a 15 ft wall as shown in the figure. What angle should the ladder make with the ground if the horizontal distance the ladder extends beyond the wall is to be a maximum? 24 Find the volume of the solid formed by rotating the region under the curve 0 ≤ x ≤ π, about the xaxis. 25 Find the volume of the solid generated by rotating the region under the curve y = tan x, 0 ≤ x ≤ π/4, about the xaxis. In Problems 3134, sketch the given function in (a) rectangular coordinates, (b) polar coordinates. Let 0 ≤ θ ≤ 2π. 35 Find the area of the polar region bounded by r = 1 + sin^{2} θ. 36 Find the area of the polar region bounded by r  sin θ + cos θ. 37 Find the area of the polar region inside both the curves r = 1  cos θ and r = 1 + cos θ. 38 Find the length in polar coordinates of the curve r = sin^{4}(¼θ), 0 ≤ θ ≤ π. 39 Find the surface area generated by rotating the polar curve r = 1  cos θ, 0 ≤ θ ≤ 7π/2, about the xaxis. 40 Use the Intermediate Value Theorem to prove that arctan y has domain ( ∞, ∞). 41 Use the Intermediate Value Theorem to prove that the domain of arcsec y is the set of all y such that y ≤  1 or y ≥ 1. 42 Prove that if/ is a differentiable function of x then 43 If u and v are differentiable functions of x then 44 Show that if f' and g are differentiable for all x then 45 Use integration by parts to prove the reduction formula Hint: 46 Suppose y = f(x), a < x < b and x = g(y), c < y < d are inverse functions and are strictly increasing. Let y_{0} = f(x_{0}). Prove that: (a) If f is continuous at x_{0}, g is continuous at y_{0}. (b) If f' (x_{0}) exists and f'(x_{0}) ≠ 0, then g'(y_{0}) exists. 47 Justify the following formula for the area of the polar region bounded by the continuous curves θ = f(r), θ = g(r), a ≤ r ≤b, where 0 ≤ f(r) ≤ g(r) ≤ 2π. 48 Justify the following formula for the mass of an object in the polar region 0 ≤ r ≤ f(θ), a ≤ θ ≤b, with density ρ(θ) per unit area. 49 Justify the following formulas for the centroid of the polar region 0 ≤ r ≤ f(θ), a ≤ θ ≤ b. Hint: The centroid of a triangle is located on a median ⅔ of the way from a vertex to the opposite side. 50 Find the centroid of the sector 0 ≤ r ≤ c, a ≤ θ ≤ b. 51 Find the centroid of the region bounded by the cardioid r = 1 + cos θ.


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