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Steady states If a function f represents a system that varies in time, the existence of
72. The amount of drug (in milligrams) in the blood after an IV tube is inserted is m(t) = 200(1 − 2−t).
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- Option D: Lim t -> 0 f(t,0) = 1 but lim t -> 0 f(0,t) = -1arrow_forwardlim y approaches 0 ................... [5y3 + 8y2]/[3y4 - 16y2]arrow_forwardevaluate (a) lim x→ x^2 - 36/x^2 + 5x - 6 (b) the demand curve of a firm is p= -10q + 5900 and its average cost is A(q) = 2q^2 - 4q + 140 + 845/q. where q is the firm's output produced and sold. 1. derive an expression for the total revenue of the firm. 2. derive an expression for the firm's total profit function 3. derive an expression for the rate of change of profit function of the firm. 4. is the rate of change of profit increasing or decreasing when the firm's output level is q= 50? 5. determine the level of output for which the total profit of the firm is maximized 6. what is the firm's maximum profitarrow_forward
- 12. Consider that the inflation rate in a six-year period in Mexico behaves according to the function i(T)=t^2 - t + 3Calculate the minimum value of inflation and the time in which it reaches this value. Consider the domain of the function [0,6] and the time in years. Provide the graphic behavior of inflation during the six-year period.In what interval does I(+) decrease? That is, there is deflation.arrow_forwardShow that the function f(x,y)=8x^2 y subject to 3x−y=9 does not have an absolute minimum or maximum. (Hint: Solve the constraint for y and substitute into f.) Solve the constraint for y. y = ? Substitute into f. f(x,y)= ? Determine the behavior of f as x approaches −∞. limx→−∞f(x,y)= ? Determine the behavior of f as x approaches ∞. limx→∞f(x,y)= ? Does this show that f does not have an absolute maximum or minimum? 1. No 2. Yesarrow_forward[3 - x x 2 let fx)=+1 a) Find lim fx) and lim f) X-2 X-2+ exist? why? Sx) b) Dose lim X2arrow_forward
- In an economic enterprise, the total amount T that is produced is a function of the amount n of a given input used in the process of production. For example, the yield of a crop depends on the amount of fertilizer used, and the number of widgets manufactured depends on the number of workers. Because of the law of diminishing returns, a graph for T commonly has an inflection point followed by a maximum, so a cubic model may be appropriate. In this exercise we use the model shown below, with n measured in thousands of units of input and T measured in thousands of units of product. T = −2n3 + 3n2 + n (a) Make a graph of T as a function of n. Include values of n up to 1.5 thousand units. (b) Express using functional notation the amount produced if the input is 1.02 thousand units. (Round your answer to two decimal places.)T( )Calculate that value. (Round your answer to two decimal places.) thousand units(c) Find the approximate location of the inflection point. (Round…arrow_forward(a) Find the following: i. Lim →x/ + 9x/ 6x/ − 4x/ii. Lim →(2x − 1)/ − 3 x − 5(b) The total revenue curve of a firm is R(q) = 40q − 12q and its average cost A(q) =1 30q − 12.85q + 20 +400 q,where q is the firms output. i. Derive an expression C(q) for the firms total cost function. ii. Derive an expression Π(q) for the firms profit function. iii. Is the rate of change of profit increasing or decreasing when the ouput level of the firm is 10 units? iv. Determine the level of output for which the firms profit is maximized. v. What is the firmss maximum profit?arrow_forwardGuess the value of the limit (if it exists) by evaluating the function at the given numbers. (It is suggested that you report answers accurate to at least six decimal places.) Let f(x)=e^1.9x−e^3.1x/x. We want to find the limit limx→0 e^1.9x−e^3.1x/x.Start by calculating the values of the function for the inputs listed in this table. x x f(x)f(x) 0.2 0.1 0.05 0.01 0.001 0.0001 0.00001 Based on the values in this table, it appears limx→0 e^1.9x−e^3.1x/x=arrow_forward
- The cost C (in dollars) of producing x units of a product is C = 1.60x + 9,000. (a) Find the average cost function C. C = (b) Find C when x = 1,000 and when x = 10,000. C(1,000) =$ per unit C(10,000) =$ per unit (c) Determine the limit of the average cost function as x approaches infinity. lim x→∞ C(x) = Interpret the limit in the context of the problem. As more and more units are produced, the average cost per unit (in dollars) will approach $arrow_forwardlim g(f(x))x->-2arrow_forwardThe size of an undisturbed fish population has been modeled by the formula pn+1 = bpn / (a + pn) where Pn is the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p0 > 0. a) If {pn} is convergent, then the only possible values for its limit are 0 andb −a. Justify. b) Given that pn+1<(b/a)pn is true. justify. c) Use part (b) to justify if a>b, then lim pn =0. where limit n is approach to infinityarrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning