M 4086,) HOMEWORK 2 You should show all of your work unless instructed otherwise. Correct answers without appropriate supporting work will receive no credit. Problem 1. Let T(u) = 3&3 211.2 + 1. (a) Compute T'(u) and T'(2). (b) Determine all solutions of the equation T'(u) = 0. (c) Determine the intervals on which T is increasing. (d) Determine the intervals on which T is decreasing. ) (e Identify the numbers at which T has a local maximum or minimum. Specify which type of extremum occurs at each number you give, justifying your answers. (Hint: Keep in mind that, if a function F is continuous (polynomials are always continuous) on an interval I, and is never zero on I, then it is either always positive or always negative on I. The sign can be determined by checking the value of F at a single number in I) Problem 2. Let f(r) : $3\". (a) Compute f'($) and f'(4)_ (b) Give the equation of the tangent line to the graph of f at a = 4 in point-slope form. (c) Use linear approximation based at a = 4 to estimate the numbers (3.1)3/2 and 53/2. Do not round your answers. (d) Compare your estimates from (c) with the values returned when you compute (3.1) and 53/2 using a calculator (or MATLAB, etc); are they overestimates or underesti mates? (e) Plot the graph of f and the tangent line at a = 4 on a common set of axes in MATLAB, and use the resulting gure to explain your answer for (d) geometrically.1 (You may have to play with the viewing window to see what is going on.) 3/2 Problem 3. Rick Sanchez has constructed a rudimentary rocket car out of stuff he found in the garage (because he can). Rick and Morty begin to travel in a perfectly straight line from the Smith family residence on a test drive of the rocket car. Rick uses science to determine that, at any given time t between t = 0 minutes and t = 6 minutes, the distance d(t) between the Smith house and the rocket car, measured in miles, is given by d(l) = 3t2 + 18b (Rick does not even try to explain his calculations to Morty.) (a) The velocity v(t] of the rocket car at time t is, by denition, the rate of change per unit time of the distance between the Smith house and the rocket car. Give a formula expressing 1.!(t) explicitly in terms of 1. What are the units of v(t)? (b) Two minutes into the drive, at how many miles per hour is the rocket car traveling? (c) What is the velocity of the rocket car ve minutes into the drive? (Don't forget units!) Is the rocket car still moving away from the Smith house, or is it moving towards it? Explain your answer. (d) At what time, between 0 and 6 minutes, is the rocket car farthest away from the Smith house? Explain your answer. Consider a company (such as Buc-ee's or Hedgehog Brewing) which produces a product (such as beaver-based confections or craft beer). The company's cost function 0 gives the cost C(13) of producing x units of their product (in the appropriate currency, such as blemarcks or US. dollars); the average cost per unit at a production level of as units, denoted C(32), is then equal to C (1:) / z. The marginal! cost at a production level of :1: units is dened to be C" (2:), the derivative of the cost function; marginal cost is the per-unit rate of change of (total) cost, and helps estimate the change in cost resulting from a (not too dramatic) change in production. The inverse demand function (or pricing function) 10 gives the highest price p(:r) the company can charge per unit while still generating the demand required to sell m units of their product. The revenue function R gives the revenue 13(3) the company generates from selling m units of their product, and is expressed in terms of the inverse demand function by the formula R(:r) = :rp(x) (number of units sold times price per unit); as in the case of the cost function, we dene average revenue and marginal revenue at a production level of as units to be E(I) = R(:c)/:r and R'(a:), respectively. Finally, the prot function P gives the prot P(:r) brought in from selling 1: units of their product; because prot is the difference between revenue and cost, P(a:) = R(a:) 0(a); the average prot at a production level of 2: units is Eb) = P(r)/x, and the marginal prot at a production level of :1: units is P' (:13) Problem 4. Economists sometimes describe the marginal cost C'(:r) as the change in cost resulting from a one-unit increase in production beyond the current level. Suppose a company has estimated that the cost (in dollars) of producing 3: units of their product is given by C(32) = 10000 + 5x + 0.25.12. (a) Compute the marginal cost at a production level of 400 units. (b) Compute the change in cost resulting from an increase in production to 401 units. (c) Are the economists wrong? Why do they say what they do? Explain your answer. Problem 5. Hedgehog Brewing has begun to brew its rst ever collaboration beer. In order to shed its wholesome image as a family brewer and expand its customer base to include fans of Metallica, etc., Hedgehog is working together with 3 Floyds Brewing (an esteemed brewer in Munster, Indiana, that can reasonably be described as very metal) to produce a new imperial stout called Gazing Into the Abyss. The hedgehog actuaries have determined that the cost to Hedgehog of producing :5 bottles of Gazing Into the Abyss is given by C (:12) = 4000 + 33: (dollars), while the price per bottle at production level n is given by p(.v) : 5 % (dollars). Hedgehog is currently brewing 3500 bottles of Gazing Into the Abyss per montlh. (a) Compute the average cost per bottle and the price per bottle at the current production level. Include units. (b) Compute the monthly cost and monthly revenue at the current production level. Include units. (c) Is the brewing of Gazing Into the Abyss currently protable to Hedgehog? Explain your answer. (d) Compute the marginal cost, the marginal revenue, and the marginal prot at the current production level. Include units. (e) Should Hedgehog increase or decrease production of Gazing Into the Abyss? Explain your answer using what you found in (d) Problem 6. Compute the derivatives of the functions given by the following formulas. (You will need the product and/ or quotient rule for this problem.) (a) m) = e. tantt) (b) Ge) = 9.2}: l' Problem 7. Compute the derivative; of the functions given by the following formulas. (You will need the chain rule for this problem.) (a) 3(2) = tan (1) (b) $905) : sinr"(t3 + t + 1) (c) 30(33) = (22:3 5:):2 9)'1 Problem 8. Here is a beautiful table of values for differentiable functions f and g, and their derivatives. Use the information in the table to compute the following derivatives: (3) (f 29W) ('0) (f9)'(3) (C) (f og)'{1) (d) (9 o f)'(1)