Need help with calculus. Thank you so much!

1. [10 points] A car is traveling on a long straight road. The driver suddenly realizes that there is a stop sign exactly 40 feet in front of the car and immediately hits the brakes. The car's velocity decreases for the next two seconds as the car slows to a stop. Let v(t) be the velocity of the car, in feet per second, t seconds after the driver hits the brakes. Some values of the function v are shown in the table below. 10 0.5 1 1.5 2 v(t) 40 32 23 12 0 a. [2 points] Estimate the car's acceleration 0.5 seconds after the driver hits the brakes. Show your work and include units. b. (5 points] Using the information given in the table, find a linear approximation L(t) for v(t) near t = 1. Answer: L(t) = c. (3 points] Suppose we used the linear approximation in part b. to approximate v(1.25). Based on the information given in the table, would this be an overestimate or underestimate or is it impossible to tell? Be sure to JUSTIFY your answer.2. [9 points] Stringer Bell is starting a fruit punch company. He has determined that the total cost, in dollars, for him to produce q gallons of fruit punch can be modeled by C(q) = 100 + 4 + 259/100 Since he is selling an clastic product in an inclastic marketplace, Stringer can sell up to 100 gallons to Avon at a price of $4 per gallon, and he can sell the rest to Snoop at a price of $3 per gallon. Assume that Stringer sells all of the fruit punch that he produces. Note: the quantities of fruit punch produced and sold do not have to be whole numbers of gallons. a. [4 points] For what quantities of fruit punch sold would Stringer's marginal revenue equal his marginal cost? Answer: b. (5 points] Assuming that Stringer can produce at most 200 gallons of fruit punch, how much fruit punch should be produce in order to maximize his profit, and what would that maximum profit be? You must use calculus to find and justify your answer. Be sure to provide enough evidence to justify your answer fully. Answer: gallons of fruit punch: and max profit:3. [9 points] The Arbor Transit Authorities (ATA) are designing rain shelters for their bus stops. They decide to place a roof in the shape of half a cylinder on four vertical legs of height y feet. The four legs are placed in a rectangle on the ground with width r fect and length y feet. The costs of production are: . $25 for each 1 foot of the total length of the legs, . $40 for each square foot of the area of the roof. The following formulas may be useful in this problem: . the surface area of a cylinder of radius r and length is 2are, . the volume of a cylinder of radius r and length ( is The ATA would like to spend exactly $1000 on one shelter. a. [5 points] Find a formula for y in terms of z. Answer: = b. |4 points] Suppose we used the above to find a formula for the volume of the covered shelter, denoted by V(x). If the ATA wants to make sure that each of the sides of the rectangle has length at least 5 feet, and the height (that is, y) of the shelter is at least 8 feet. What is the domain of the function V(x)? Answer:4. [10 points] In the following questions, use calculus to justify your answers and show enough evidence to demonstrate that you have found them all. Determine your answers algebraically. a. [6 points] Let /(r) be a continuous function defined for all real numbers whose derivative is given by M(I) = (2r + 1)(x - 2)2 (x + 3)1/3 Find the r-coordinate(s) of all local extrema of the function /(r). Write "NONE" if the function has no local extrema. b. |4 points] Suppose now that we consider /(x) on the domain (3, co). Given that /(3) = -5, determine the r-coordinates of any GLOBAL extrema of /(r). II none exist write "NONE". Be sure to justify your answers.5. [13 points| A snowman was built on the diag, and since it is a bright and sunny day his head starts to melt. The surface area of his head (which is a perfect sphere) decreases at a constant rate of 40 in* per minute. Recall that the surface area of a sphere of radius r is S = 4ar and the volume is V = = ap. Be sure to include unils in your answers. a. [3 points] How fast is the radius of the snowball changing when the radius is 5 inches? Answer: b. |4 points] How fast is the volume changing when the radius is 5 inches? Answer: c. [3 points] Write a formula for V in terms of S and r. Your expression should include both S and r. Answer: V = d. [3 points] Use the above formula to verify the rate at which the surface area is changing when the radius is 5 inches. You should use your answer from parts a. and b