d. ATC'(Q) for the average total cost function ATC(Q) = Q2 - 75Q + 2000 + 2000 e. f' (x) for the function f (x) = (axb+ cxd)w (exf+gach) 3. Using the chain rule, find expressions for a. f'(x) for the function f (x) = (x2+ 1)99 b. f'(x) for the function f (x) = (4) c. f' (x) for the function f(x) = Vx2 + 1d- P'(Q) for the inverse demand function P(Q) = #100 20 Activity 2: Marginal Analysis for a Perfectly Competitive Firm 4. In the model of perfect competition [a type of market structure], all rms are assumed to be price-takers; and thus choose only output to maximize prot Let the tire market be perfectly competitive and a representative rm in this industry have the following price and total cost functions: F = 100 TC(Q) = 50 + 600 + so2 where P is the price, TC (Q) is the total cost, and Q is output of the rm in hundreds of tires per month. a. Specify the total revenue function, TR(Q). with 'vinsmt (1. Sketch the marginal revenue and marginal cost curves on the same graph. e. Using marginal analysis, what is the optimal output? Call this (2* and show it on your graph. f. What is the price level at this output? Call it P' and show it on your graph. meade Activity 3: Using Marginal Analysis to Sketch a Curve 5. Give a rough sketch of the following curve (on the next page) f(x) =-x3+3x a. Find the y-intercept b. Take the first and second derivative, f' (x) and f" (x) c. Set the first derivative equal to zero, and solve for the values of x that satisfy this equation. Explain, in words, what these two points tell us about the function F (x). d. Plug these zero points from part c. into the second derivative. What does the value of the second derivative tell us about f (x) at these two points? e. Plug the two values of x from part c. into the original function, f (x), and make a rough sketch of f (x)