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Math 125 Carter Take Home Quiz Instructions. This quiz is singular. Take it home this weekend, and solve each problem. I will grade each problem on an “all correct” or not scale: 1 or 0. These problems are edited and culled from the last two finals that I gave. They are not indicative of the entire final, but a substantial subset thereof. So you should work these to begin studying for the final. Now is a good time to start. 1. The area of a circle is increasing at the rate of 10 square centimeters per second. How fast is the radius increasing when the radius is 10 centimeters? (The area of a circle is A = πr2 ). 2. Coffee drips at a rate of 5 cubic centimeters per second from a 18 centimeter tall Melita cone whose radius is 9 centimeters. How fast is the height of the coffee decreasing when the height is 5 centimeters? The volume of a cone is V = π3 r2 h. 3. A rectangle is inscribed with two vertices on the parabola y = 48 − x2 and with its base along the x-axis. What are the dimensions of the rectangle of maximal area that fits this description? 4. Sketch the graphs of the following functions. (a) f (x) = 3x2 + 5x − 17 (b) f (x) = x2 (x − 4) (c) f (x) = x−1 x2 + x f (x) = x2 + x x−1 f (x) = x−1 x+1 (d) (e) (f) f (x) = 2 sin (3x − π) for x ∈ [0, 2π]. 1 (g) f (x) = 3 sin (2x) for x ∈ [−2π, 2π]. 5. (5 points each) Compute the following anti-derivatives and definite integrals (a) 4 Z x3 dx 1 (b) Z (x + 1) √ dx x (c) Z dx 2x (d) Z e2x dx (e) π/2 Z cos (x)dx −π/2 (f) 5 Z (3x + 7) dx 2 (g) Z sec2 (x) dx (h) Z π/2 cos (x) dx −π/2 (i) Z 2 e10x dx