Prof. Ming Gu, 861 Evans, tel: 2-3145Email: [email protected]://www.math.berkeley.edu/∼mgu/MA16AMath16A Sample Midterm II Solutions, Fall2009This is a closed book, closed notes exam. You need to justify every one of youranswers unless you are asked not to do so. Completely correct answers given withoutjustification will receive little credit. Look over the whole exam to find problems thatyou can do quickly. You need not simplify your answers unless you are specificallyasked to do so. Hand in this exam before you leave.Problem Maximum Score Your Score1 202 203 204 205 106 10Total 100Your Name & SID:Your Section & GSI:Math16A Sample Midterm II Solutions, Fall 2009 21. Determine the derivatives of the following functions(a) f(x) =x2− 1x2+ 1.Solution: f0(x) =4x(x2+1)2.(b) f(x) =x2+ 1√x2− 1.Solution: f0(x) =x(3x2−1)√x2−1.Math16A Sample Midterm II Solutions, Fall 2009 32. Determine the derivativesdydxof the following functions at the given points(a) y(x) as a function of x defined by the equationx3+ y3= 2,at point (1, 1).Solution:dydx= −x2y2= −1.(b) y(x) as a function of x defined by the equationxy3+ x2y2+ x3y = −1,at point (−1, 1).Solution:dydx= −y3+ 2xy2+ 3x2y3xy2+ 2x2y + x3= 1.Math16A Sample Midterm II Solutions, Fall 2009 43. Consider the composition of three functions(a) Let f(x), g(x), and h(x) be differentiable functions. Derive a formula for the derivativeof the function f(g(h(x))).Solution:ddx(f(g(h(x)))) = f0(g(h(x)))g0(h(x))h0(x).(b) Write the functionq1 +√1 + x2in the form f(g(h(x))), and use your formula fromabove to compute the derivative of this function.Solution: Let f(x) =√1 + x, g(x) =√1 + x and h(x) = x2.ddx(f(g(h(x)))) =x2√1 + x2q1 +√1 + x2.Math16A Sample Midterm II Solutions, Fall 2009 54. Let f (x) = (2x2+ 3)3/2. Show that f(x) is decreasing for x < 0 and increasing for x > 0.Sketch the graph of this function and find the global minimum of f(x) for −∞ < x < ∞.Solution:f0(x) = 6x√2x2+ 3, f00(x) =6(4x2+ 3)√2x2+ 3.The factor√2x2+ 3 in f0(x) is always positive. Hence f0(x) < 0 is negative for x < 0 andpositive for x > 0. Therefore f (x) is decreasing for x < 0 and increasing for x > 0. Sincef00(x) is always positive, the graph is concave up, without inflection points. The minimumis at x = 0.Math16A Sample Midterm II Solutions, Fall 2009 65. If the demand equation for a monopolist is p = 150 −0.02x and the cost function is C(x) =10x + 300, find the value of x that maximizes the profit.Solution: The profitP (x) = xp − C(x) = x (150 − 0.02x) − (10x + 300) = 140x − 0.02x2− 300,which is a quadratic with negative second order coefficient −0.02. Hence the graph is concavedown, with max atP0(x) = 140 − 0.04x = 0,or x = 3500.Math16A Sample Midterm II Solutions, Fall 2009 76. An open rectangular box is to be 4 feet long and have a volume of 200 cubic feet. Findthe dimensions for which the amount of material needed to construct the box is as small aspossible.Solution: Let the height be x feet and width y feet. Then the constraint is 4xy = 200 orxy = 50. Since the box is open, one only needs to construct the four sides and the bottom ofthe box. Hence the objective to b e minimized is 4y + 2(xy + 4x) = 2xy + 4y + 8x. Replacey by 50/x, the objective isC(x) = 2x ∗ 50/x + 4 ∗ 50/x + 8x = 100 + 200/x + 8x.LettingC0(x) = −200/x2+ 8 = 0,we have x = 5 and hence y =