Computing gradients Compute the gradient of the following functions and evaluate it at the given point P
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- Maximum Sales Growth This is a continuation of Exercise 10. In this exercise, we determine how the sales level that gives the maximum growth rate is related to the limit on sales. Assume, as above, that the constant of proportionality is 0.3, but now suppose that sales grow to a level of 4 thousand dollars in the limit. a. Write an equation that shows the proportionality relation for G. b. On the basis of the equation from part a, make a graph of G as a function of s. c. At what sales level is the growth rate as large as possible? d. Replace the limit of 4 thousand dollars with another number, and find at what sales level the growth rate is as large as possible. What is the relationship between the limit and the sales level that gives the largest growth rate? Does this relationship change if the proportionality constant is changed? e. Use your answers in part d to explain how to determine the limit if we are given sales data showing the sales up to a point where the growth rate begins to decrease.arrow_forwardComputing gradients Find ∇ƒ(3, 2) for ƒ(x, y) = x2 + 2xy - y3.arrow_forwardComputing gradients Compute the gradient of the following functions and evaluate it at the given point P.arrow_forward
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- Chain Rule Use the Chain Rule to evaluate the following derivatives. w'(t), where w = z ln (x2 + y2), x = 3et, y = 4et, and z = tarrow_forwardEconomists who study production of goods by a firm consider two functions. The revenue function R(x) is the revenue the firm receives when x number of units are sold. The cost function C(x) is the cost the firm incurs when producing x number of units. The derivatives of these functions R′(x) and C′(x) are called by economists the marginal revenue and cost function. The figure shows graphs of the marginal revenue function R′ and the marginal cost function C′ for a manufacturer. Assume that R and C are measured in thousands of dollars.arrow_forwardUsing Leibniz find the third derivative of the following function y=x2cos3x Find the local linear approximation L(x) of the function f(x)=lnx at x=1 .arrow_forward
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