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Directional DerivativeConsider the function
and the unit
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Chapter 13 Solutions
Calculus
- 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_forwardAnalyzing critical points Use the Second Derivative Test to classify the critical points of ƒ(x, y) = xy(x - 2)(y + 3).arrow_forwardAnalyzing critical points Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points. ƒ(x, y) = yex - eyarrow_forward
- Derivative rules Suppose u and v are differentiable functions at t = 0 with u(0) = ⟨0, 1, 1⟩, u'(0) = ⟨0, 7, 1⟩, v(0) = ⟨0, 1, 1⟩, and v'(0) = ⟨1, 1, 2⟩ . Evaluate the following expression.arrow_forwardcurve sketching f(x)=x*exarrow_forwardDemand Function for Desk Lamps The demand function for the Luminar desk lamp is given by p = f(x) = −0.1x2 − 0.9x + 28 where x is the quantity demanded in thousands and p is the unit price in dollars. (a) Find f '(x). f '(x) = (b) What is the rate of change of the unit price (in dollars per 1,000 lamps) when the quantity demanded is 3,000 units (x = 3)? $ per 1,000 lamps What is the unit price (in dollars) at that level of demand? $arrow_forward
- Identifying homogenous or non homogenous functions. Show your complete solution. when x=0, y=3 xy3dx+ex^2dy=0arrow_forwardThink About It Describe the relationship betweenthe rate of change of y and the rate of change of x ineach expression. Assume all variables and derivativesare positive. (a) $\frac{d y}{d t}=3 \frac{d x}{d t}$(b) $\frac{d y}{d t}=x(L-x) \frac{d x}{d t}, \quad 0 \leq x \leq L$arrow_forwardDerivative of the function f (x) = tan x at x = n/4arrow_forward
- Derivative rules Suppose u and v are differentiable functions att = 0 with u(0) = ⟨2, 7, 0⟩, u'(0) = ⟨3, 1, 2⟩, v(0) = ⟨3, -1, 0⟩,and v'(0) = ⟨5, 0, 3⟩. Evaluate the following expressions.arrow_forwardRegression y on x And x on y .arrow_forwardAnalyzing critical points Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points.arrow_forward
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