MATH 2500: TAKE HOME 03 (30 points.) NAME: DUE: Wednesday, June 10th by 8 AM. DIRECTIONS: To receive full credit, make sure your work is neat and complete. 1. Find and simplify the indicated derivative using derivative properties: (a) \u0003 d \u0002 3 2x 5x + 1 dx (b) For F (x) = (c) For y = x sin(2x), find F 0 (x). 3x + 5 , find y 0 . 7 2x \u0002 \u0003 (d) Find D tan2 () . (e) Find y 0 for y = tan1 (4x) 10 2. Find and simplify the second derivative using derivative properties: (a) f (x) = sin1 (2x). (b) f (x) = cos(x) 1 sin(x) HINT: Be on the lookout for 'Pythagorean Magic' to help you simplify f 0 (x). 10 3. Suppose T (t) gives the temperature (in degrees Fahrenheit) on a certain day t hours after 6 AM. (a) Interpret the statement: T 0 (4) = 2 in terms of time and temperature. (b) If T (12) = 70 and T 0 (t) = 0 for 12 t 18, what does this mean in terms of time and temperature? 4. The height (in feet) of a basketball off of the ground t seconds after it is thrown in the air is given by: s(t) = 16t 2 + 24t + 8, 0 t 2. (a) Find expressions for the velocity, v (t), and acceleration, a(t). (b) Is the ball speeding up or slowing down when t = 1? Explain. 5. Suppose C (x) gives the cost, in dollars, to produce x phon-i smart phones and suppose R(x) gives the revenue, in dollars, generated by selling x phon-i smart phones. (a) Interpret the statement: MC (500) = 250 in terms of dollars and phones. (b) Interpret the statement: MR(500) = 100 in terms of dollars and phones. (c) Should the manufacturer continue to produce and sell more than 500 phon-i phones? Explain. 10