4. (15 points) Let F(x,y,z) = x² - y²z. What is the direction of greatest increases for the function F at the point (1,2,3)?

5. (20 points) Use the method of Lagrange multipliers to find the point on the plane x + 2y + 3z = 6 that is nearest to the origin.

6. (20 points) Evaluate the integralwhere C is the the right triangle with vertices (0,0), (0,2), and (1,0).

1.The demand and supply equations for an e‐book reader are given by p = 195 -3.8x q = 150 +3.2xwhere p is the price in dollars and x represents the number of units in millions. Find the equilibriumpoint for this market.

2.Find an equation of the tangent line to the graph of y= at  x=1.

3.The demand function for a product is modeled by p=1/(0.000001x3+0.01x+1) where p is measured in dollars and x is measured in thousands of units. Find the rate of change of the demand x with respect to the price p when x=100.

4.Show that the graph of the normal probability density function f(x)= has points of inflection at x=±1.

5.The marginal cost of producing x units of a product is modeled by.  It costs $50 to produce one unit. Find the total cost of producing 200 units.

6.Find the area of the region bounded by the graphs of f(x)=3x3-x2-10x and g(x)=-x2+2x.

7.A manufacturer’s production is modeled by the cobb‐Douglas function f(x,y)= Where x represents the units of labor ( at $ 150 per unit ) and y represents the units of capital ( at $ 250 per unit ). The total cost for labor and capital cannot exceed $ 50,000. Will the manufacturer’s maximum production level exceed 16,000 units? ( Hint: Use the method of Lagrange Multipliers and the funciton . F(x,y,λ)=f(x,y)-λ(150x+250y-50,000).)

8.A. Find the limit.

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