Management Calculus MATH

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Management Calculus MATH.1220 exam 2 online spring 2021

M. Stick

Show all analytic work. Graphs can be done by hand. There is a total of 105 points.

1. y = 2/5 x³ – 6x² , 0 £ x £ 15.

a) Find the (x, y) coordinates of the point of inflection.

b) Use the test for concavity and determine over what interval the revenue is concave up and where is it concave down?

c) Find the (x, y) coordinates of the critical points.

d) Check the (x, y) coordinates of the end points and find all absolute max and absolute min points. Also graph the curve and label the inflection point and critical points.

2. Cost is 4.8x+2000, price 1s 14.4 - 0.002x.

a) Find the (x, y) coordinates of the maximum profit and use the 2" derivative test to verify that you have a relative max.

b) Show all work to find coordinates of the max revenue and determine the best price to maximize the revenue?

3. A rectangular plot is bounded on three sides by fencing. Exactly 170 feet of fencing will be used to enclose the plot. Find the dimensions of the lot to maximize the total area and find the total area enclosed. Y,  x,   y

a) Define the constraint equation and objective function for this problem.

b) What are the length of the sides of the rectangular plot? Also, what is the minimum amount of fencing used? Show all work.

4. y =Ö2x+3 , x=3 and Dx=2.

a) Evaluate the differential dy accurate to 3 decimal places.

b) Evaluate the actual change Dy accurate to 3 decimal places.

c) Graph the original curve y=Ö2x+3 and superimpose on the graph the quantities dy and Dy and label each of the quantities.

5. The demand D(x) = Ö15-0.25x .

a) Find the elasticity.

b) When the price x = $48, show work to determine whether the demand is elastic or inelastic.

c) Determine the value of the maximum revenue accurate to 2 decimal places and also determine the values of the price x for which the total revenue is inelastic.

6. Revenue R(x) =100x—0.4x² and cost C(x)=10+0.5x. The units x depend on time t in days.

a) Find the rate of change in the total revenue with respect to time t and the rate of change in the total cost with respect to time t.

b) When x= 3 units and the rate of change in x with respect to time t is 2 units per day, find the rate of change in the profit with respect to time.

7. Given the curve xy² —2x³ + y = 5

a) Differentiate implicitly to find dy/dx.

b) Given a point on the curve accurate to 1 decimal place as x = 1, y = -3.2, find the slope of the curse at that point accurate to 3 decimal places. Also, graph the tangent line at that point on the adjacent graph of the curve.

8. a) Given that A = Pe^rt where the amount A = $10,000 and the original; principal P = $4,000. Time t in years is 8 years. What yearly interest rate r accurate to 3 decimal places and compounded continuously 1s required to achieve the desired result for A?

b) y= 3e^-x2 +2. Find the equation of the tangent line to this curve at x = 1. Express results accurate to 3 decimal places. Also graph the curve and superimposed tangent line at x = 1.

c) y=In x²e^-x/3-2x . Find the derivative of this expression using the properties of logarithms. The LCD is required.