1. Calculate the second derivative for the following functions. a) y = ma + ex + 2021 b) f(x) = sin? (x) c) f(x) = x + eTx d) y = In (ezz )2. The top graph shows the number of COVID cases in Johnson County, Iowa from June 6, 2020 through three graphs to answer the following questions September 28, 2021. (Source: UIOWA COVID-19 data - with some smoothing for nicer derivatives) The 200 middle and bottom graph show the first and second derivatives for the number of cases, respectively. Use all 150 Cases 100 COVID Cases in Johnson County lowa (7-day average) 50 O 20 10 First derivative 0 -10 -20 2 Second derivative N 20 GO 100 120 140 160 180 200 220 240 260 280 Days since June 6 2020 320 340 360 380 400 420 440 460 480 500(a) On day 75, the rst derivative has a value of 7.8. Explain what this means in relation to the number of COVID cases. (b) On day 153, the first derivative is positive, and the second derivative is negative. Explain what this means in relation to the number of COVID cases. Would this make you optimistic or pessimistic about the outlook of COVID at day 153? (c) On day 1?6, the rst derivative is negative, and the second derivative is positive. Explain what this means in relation to the number of COVID cases. Would this make you optimistic or pessimistic about the outlook of COVID at day 176? (d) On both days 75 and 447, there were approximately 41 active cases of COVID-19. Why is there reason for a more optimistic outlook on day 447 compared to day 75? 3. A helicopter starting on the ground is rising directly into the air at a rate of 25 ft/sec. You are running on the ground starting directly under the helicopter at a rate of 10 ft/sec. Find the rate of change of the distance between the helicopter and yourself after 5 seconds.4. A tank shaped like an inverted cone is leaking water, but you do not know at what rate. The tank has a height of 16 ft and a radius of 5 ft. If the water level is decreasing at a rate of 0.25 ft/min when the depth of the water is 8 ft, determine the rate at which water is leaking out of the cone.5. Use a linear approximation to estimate the following quantities. a) b) $8.01 0.986. I want to paint the hemispherical dome atop Dallas Hall with a coat of paint that is 11' 10 inches thick, and I need to buy some paint. Assume that the radius of the dome is 50 feet. Use differentials to estimate the amount paint (in cubic feet) that I will need to finish the project