Math 10C - Fall 2009 - Final ExamName:Student ID:Section time:Instructions:Please print your name, student ID and section time.During the test, you may not use books or telephones. You may use a ”cheat sheet” of noteswhich should be a page, front only.Read each question carefully, and show all your work. Answers with no explanation will re-ceive no credit, even if they are correct.There are 10 questions which are worth 150 points. You have 180 minutes to complete the test.Question Score Maximum1 152 203 134 175 156 157 158 159 1010 15Total 150Problem 1. [15 points]At what point (x, y, z) on the plane x + 2y − z = 5 does the minimum of the functionf(x, y, z) = x2+ 2y2+ (z + 1)2occur?Problem 2. [20 points.]Consider the functionf(x, y) = 3y2− 2y3− 3x2+ 6xy.(i) [8] Find the critical points of the function.(ii) [8] Determine the nature of the critical points (local min/local max/saddle).(iii) [4] Does the function f(x, y) have a global minimum or a global maximum?Problem 3. [13 points] Consider the functionf(x, y) = ln(xy2) −2xy(i) [8] Compute the second order Taylor polynomial of f around (1, 1).(ii) [5] Find the tangent plane to the graph of f at the point (1, 1, −2).Problem 4. [17 points.]Find the global minimum and global maximum of the functionf(x, y) = x2+ y2− 2x − 2y + 4over the closed diskx2+ y2≤ 8.Problem 5. [15 points]Consider the functionf(x, y) = 1 + x2+ y2.(i) [4] Draw the contour diagram of f labeling at least three levels of your choice.(ii) [4] Compute the gradient of f at (1, −1) and draw it on the contour diagram of part (i).(iii) [3] Does the function f have a global minimum? If no, why not? If yes, what is the mini-mum value?(iv) [4] Draw the graph of the function f.Problem 6. [15 points]Consider the functionf(x, y) = e−3x+2y√2x + 1 .(i) [5] Calculate the gradient of f at (0, 0).(ii) [5] Find the directional derivative of f at (0, 0) in the direction u =i+j√2.(iii) [5] What is the unit direction for which the rate of increase of f at (0, 0) is maximal?Problem 7. [15 points]Consider the planesx + 2y − z = 1, x + 4y − 2z = 3.(i) [4] Find normal vectors to the two planes.(ii) [6] Are the two planes parallel? Are they perpendicular?(iii) [5] Find a vector parallel to the line of intersection of the two planes.Problem 8. [15 points.]Consider the functionw = u2v e−vand assume thatu = x2− 2xy, v = −x + 2 ln y.Calculate the values of the derivatives∂w∂xand∂w∂yat the point (x, y) = (1, 1).Problem 9. [10 points]You deposit $1, 000 into your savings account every year for the next 10 years. You make thefirst deposit on January 1, 2010, and the last deposit on January 1, 2019. The interest rate for theaccount is r = 10% per year. How much money will there be in your account at the end of the 10thyear, on December 31, 2019?Express your answer in the simplest closed form. You don’t need to evaluate the powers thatmay appear in the final expression.Problem 10. [15 points]The outcome x of a certain experiment has values between 0 andπ2, with probability distributionfunctionp(x) = sin x, for 0 ≤ x ≤π2.(i) [4] Calculate the cumulative distribution function.(ii) [4] Calculate the median outcome of the experiment.(iii) [7] Possibly using integration by parts, calculate the mean outcome of the
View Full Document