Math 21a second practicy exam for Hourly 2What follows is a model of the second hourly exam (it is more or less the exam given lastfall in this course). The upcoming exam on Wednesday, November 15 will be roughly similar tothis one in length and difficulty. To study for the upcoming exam, first work the suggestedproblems for the Chapters 2, 3.1-3.2 and 4.4 and in the Supplement on Lagrange Multipliers. Theexam will cover solely material from these readings. The list of suggested problems is posted onthe course website. You should also work the practice problems for Hourly 2 supplied elsewhereat this website. After you have worked the suggested problems and checked your answers withthose in the book, and worked the practice problems and checked your answers, try taking thisexam as practice for the real thing. In this regard, note that you will have two hours for the realthing, but we hope that most people finish under the two hour limit. The answers to this practicehourly are provided elswhere at the website.By the way, in the real exam, each problem will be printed on a separate page and youwill be asked to provide your answers on that page and to use the back of that page if you needmore space to show your work.Remember that the exam on Wednesday, November 14 is from 7:30-9:30pm in LectureHalls C and E of the Science Center. Please come a few minutes early as the exam will startpromptly at 7:30pm.1) __ 2) __ 3) __ 4) __ 5) __ 6) __ : Total ______Name: _____________________________________________Circle the name of your Section TA:Arinkin(10)•Arinkin(12)•Bamberg•Cornut•Kaplan•Karu•Knill•Libine•Liu•Taubes•WilliamsInstructions:• Print your name in the line above and circle the name of your section TF.• Answer each of the questions below in the space provided. If more space is needed, use theback of the facing page.• Please give justification for answers if you are not told otherwise.• Please write neatly. Answers which are deemed illegible by the grader will not receivecredit.• No calculators, computers or other electronic aids are allowed; nor are you allowed to refer toany written notes or source material; nor are you allowed to communicate with otherstudents. Use only your brain and a pencil.• You have 2 hours to complete your work.• Note that vectors are indicated below by bold face type.In agreeing to take this exam, you are implicitly agreeing to act with fairness and honesty.1. Suppose that f(x, y) is a function with the following properties:• f(1, 2) = 6.• The directional derivative at (1, 2) of f in the direction of u = (35,45) is 5• The directional derivative at (1, 2) of f in the direction of v = (-12,12) is 12.Given this information, answer the following questions:a) Write the gradient of f at (1, 2). b) Write an equation for the tangent plane to the graph of f at the point (1, 2, 6).c) Estimate the value of f(1.02, 2.05).2. Let a, b and c be non-negative constants and F the functionF(x, y, z) = - x ln(x) - y ln(y) - z ln(z) - a x - b y - c z.Find the stationary points of F where x > 0, y > 0, z > 0 and x + y + z = 1. Don’t search forstationary points where any of x, y or z are zero.(If you are interested in where this question comes from, here is the answer: This sort ofextremal problem arises when predicting the shapes of biological molecules. For example,proteins are complicated molecules that often fold into a number of stable shapes. Moreover,the biological effect of the protein is very sensitive to the particulars of its shape. If a certainprotein has three possible folded shapes which we label as X, Y and Z, then in a givenenvironment, a fraction, x of the protein molecules will have shape X, a fraction y will haveshape Y and a fraction z will have shape Z. It is a fundamental problem in biochemistry topredict these fractions x, y and z. In particular, these fractions are known to depend on theenergies, a, b and c of the shapes X, Y and Z; and the correct values for x, y and z are theconstrained minima of F(x, y, z) = -x ln(x) - y ln(y) - z ln(z) - a x - b y - c z where theconstraint is x + y + z = 1. In this context, the function F is called the free energy.)3. Integrate the function f(x, y) = x over the interior of the region where x ≥ 0 and x2/4 + y2 ≤ 1.4. Let f(x, y) = x4 – 2 x2 + 3 y3 – 9 y.a) Find all stationary points of f.b) Classify each of the points found in Part a as local maxima, minima or saddle.c) Find all points on the level set f = 2 where the tangent line is parallel to the y axis. 5. Evaluate dy dxArc x001 sin( )∫∫() .6. A rectangular, open top aquarium with a slate bottom and glass side is to hold 20,000,000cubic centimeters of water. Suppose that glass cost 10 cents per square centimeter and slatecost 50 cents per square centimeter. Find the dimensions (height, length and width) whichminimize the price of the aquarium. Please justify your
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