Unformatted text preview: APPM 2350 EXAM 2 FALL 2010 INSTRUCTIONS: Electronic devices, books, and crib sheets are not permitted. Write your (1) name, (2) instructor’s name, and (3) recitation number on the front of your bluebook. Work all problems. Show your work clearly. Note that a correct answer with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work may receive partial credit. 1. (25 points) Consider a surface with height h described by the function h(x, y ) = x e−(x (a) Determine the location of all critical points of h(x, y ). (b) Classify all critical points. (c) What are the local extreme values of h? (d) If you were to stand on the surface at the point P corresponding to x = 0 and y = 0, in which direction(s) would you walk to increase your elevation most rapidly? What would the rate of increase be? (e) Again at point P , in which direction(s) would you travel to keep your elevation constant? 2. (25 points) Suppose that you are doing some consulting for a car company that manufactures three colors of cars, black, blue, and red. Let x, y , and z denote the number of black, blue, and red cars manufactured. After doing some research, you have determined that the total sales the company brings in each year in dollars is approximated well by the function xyz 2 , and it costs the company $10,000 to make each car, no matter what the color is. The factory in which the cars are produced can manufacture 8000 cars per year, and all vehicles that are manufactured are sold. (a) Determine the equation for the proﬁt that the company makes in a year. Recall that the proﬁt is given by the total sales minus the expenses. (b) Write down the equation that constrains the amount of proﬁt that the company can make in a year. (c) Determine the number of black, blue, and red cars that the company should manufacture to maximize their proﬁts. 3. (25 points) Alvin the Ant is walking across a metal plate that is two units wide in the xdirection and four units tall in the y direction. The bottom left corner of the plate is located at (0, 0). As he walks, he follows the path described by r(t) = t i + t2 j for t ≥ 0. The temperature of the plate is described by the function T (x, y ) = xy . (a) When, and where, does Alvin see the maximum value of dT /dt? What is the maximum value of dT /dt? (b) Let s be the distance traveled by Alvin as he walks along the plate. When, and where, does Alvin see the maximum value of dT /ds? What is the maximum value of dT /ds? (c) Now, suppose Alvin walks twice as fast along the same path. Will either of the maximum values of dT /dt or dT /ds change? If a value changes, clearly state by how much it changes and support your answer. 4. (25 points) Consider the function f (x, y ) = x3 y 3 + x2 y 2 . (a) Calculate the second order Taylor approximation to f (x, y ) near the point (1, 1). (b) Use your result from part (a) to estimate the value of f (1.1, 1.1). (c) Calculate an “upper bound on the error” associated with this second order approximation assuming that you only use values of x and y such that x − 1 ≤ 0.1 and y − 1 ≤ 0.1.
2 +y 2 ) . Projections and distances projA B = Arc length, frenet formulas, and tangential and normal acceleration components ds = v dt dT = κN ds dB = −τ N ds T= dr v = ds v  ds N= dT/ds dT/dt = dT/ds dT/dt A·B A A·A − → P S × v  d= v  − P→ · n d= S  n
B=T×N τ =− dB ·N ds dv  a N = κ  v  2 =  a 2 − a 2 T dt Directional derivative, discriminant, and Lagrange multipliers
a = aN N + aT T aT = df = ( ∇f ) · u ds Taylor’s formula (at the point (x0 , y0 )) f (x, y ) = f ( x 0 , y 0 ) + ( x − x 0 ) f x ( x 0 , y 0 ) + (y − y 0 ) f y ( x 0 , y 0 ) + fxx fyy − (fxy )2 ∇f = λ∇g, g=0 κ= d T  v × a = = 3
v  f (x) xy − y x ˙ ¨ ˙¨ =2  1 + (f (x))2 3/2  x + y 2  3/ 2 ˙ ˙ 1 (x − x0 )2 fxx (x0 , y0 ) + 2(x − x0 )(y − y0 )fxy (x0 , y0 ) + (y − y0 )2 fyy (x0 , y0 ) 2! 1 + (x − x0 )3 fxxx (x0 , y0 ) + 3(x − x0 )2 (y − y0 )fxxy (x0 , y0 ) 3! + 3(x − x0 )(y − y0 )2 fxyy (x0 , y0 ) + (y − y0 )3 fyyy (x0 , y0 ) + · · · Linear approximation error E (x, y ) ≤ 1 M (  x − x 0  +  y − y0  ) 2 , 2 where max{ fxx , fxy , fyy  } ≤ M ...
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 Fall '07
 ADAMNORRIS
 Numerical Analysis, Approximation, Automobile, Walking, x0, dT /dt

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