Math Example Set 12A. m10360/homework.html

Transcription

1 Estimating Partial Derivatives. Math Example Set 12A You should review the estimation for derivative for one variable functions. You can find a pdf pencast that reviews the backward, central, and forward difference formulae for estimating derivative at the usual website: See row for this week. m10360/homework.html 1. The temperature (F ) adjusted for wind-chill is a temperature which tells you how cold it feels, as a result of the combination of wind speed (W ) and temperature (T ). So we have F (T, W ). (Source: Wikipedia) Wind Speed (W mph) Temperature (T F) 1a. Estimate the rate of change of F with respect to T at T = 20 and W = 15? In another words, estimate the rate of change of F in the T direction at (20, 15). You should compute as many estimates as the data allows. 1b. Estimate the rate of change of F with respect to W at T = 5 and W = 25? In another words, estimate the rate of change of F in the W direction at ( 5, 25). You should compute as many estimates as the data allows. Section 14.6 (Chain Rule) Online Text. 2. Find formulas for the following derivatives by first drawing a tree diagram to connect all related quantities: 2a. 2b. du dt where u = ln(x2 + y 2 ); x = cos 2t and y = sin t. u t and u s where u = ex 1+4x 2 x 3 ; x 1 = 2t s, x 2 = t 2 and x 3 = t + 3s. Implicit Differentiation 3. Find z x = z x and z y = z in each of the following expressions: a. x 3 yz = y 2 + 3x 2xz 3 7. Find also z y (1, 1, 1). b. e yz + 1 x + 2z = 4 3. Find also z x(1, 0, 1). 1

2 Math Example Set 12B Section 14.6 (Chain Rule) Online Text. Topic: Applications of Chain Rule Linear Approximation of Change in a Function. Section 14.4 (Pg 808) Online Text. a. Consider a particle moving from point (a, b) to point (a+k, b+h). If the particle travel at a constant speed and the total duration of the motion is 1 second, find in terms of time t (in seconds), a formula for the position (x, y). b. Consider a smooth function f(x, y). Then all its partial derivatives exist and are continuous for all points near (a, b). If (x, y) is a point on the line segment in Q2(a), find a formula for the rate of change of f with respect to t. c. For a small change in time t, let the corresponding change in x from a be x, the corresponding change in y from b be y, and f be the corresponding change in f from f(a, b). Then we have f t df dt. t=0 Prove that f (a, b) x + (a, b) y x where f = f(a + x, b + y) f(a, b). This boxed formula is called the Linear Approximation of change in f when (x, y) changes from (a, b) to (a + x, b + y). 1. Using linear approximation, estimate the change in g(x, y) = xe x2y when (x, y) changes from (1, 0) to (0.9, 0.2). That is estimate the value g(0.9, 0.2) g(1, 0). 2

3 Sensitivity and Elasticity: Let z = f(x, y). Set x = a, y = b. Then the sensitivity of the quantity z relative to x at (a, b) is measured by (a, b). x We call this the sensitivity coefficient of f with respect to x at (a, b). Likewise, the sensitivity coefficient of f with respect to y at (a, b) is (a, b). On the other hand, the elasticity of the quantity z relative to x is the percentage change in z given a 1% change in x from x = a (with no change in y = b). This percentage change in z is also called elasticity coefficient with respect to x at (a, b). Likewise, the elasticity of the quantity z relative to y is the percentage change in z given a 1% change in y from y = b (with no change in x = a). This percentage change in z is also called elasticity coefficient with respect to y at (a, b). Let z = f(x 1, x 2,..., x n ). Set x 1 = a 1, x 2 = a 2,... x n = a n. Then the sensitivity of the quantity z relative to x i at (a 1, a 2,..., a n ) is measured by x i (a 1, a 2,..., a n ). We call this the sensitivity coefficient of f with respect to x i. On the other hand, the elasticity of the quantity z relative to x i is a measurement of (not exactly equal to) the percentage change in z given a 1% change in x i from x i = a i (with no change in the other variable). This percentage change in z is also called elasticity coefficient with respect to x i at (a 1, a 2,..., a n ). In general the elasticity coefficients of z are dependent on the independent values x 1, x 2,..., x n. 2. Consider a cylindrical rod with height (length) 100cm and diameter 5cm. 2a. If the measuring instrument has an error of 0.1 cm, estimate using linear approximation the corresponding error in the value of the volume if the above measurements are used. 2b. What is sensitivity of the volume of the given rod to its (i) height and (ii) diameter? That is computer the sensitivity coefficients (first partial derivatives) with respect to the height and (independently) the diameter at (100, 5). 2c. Discuss the elasticity (proportional sensitivity) of the volume relative to its dimensions. That is discuss how the percentage change in volume for a 1% change in the height (alone) compared to the percentage change in volume for a 1% change in the diameter (alone). 3. A boundary stripe 3 inches wide is painted around a rectangle whose dimensions are 100ft by 200ft. Use linear approximation to estimate the number of square feet of paint in the stripe. 3

4 Math Example Set 12C Section 14.8 Topic: Optimization with a Constraint Using Lagrange Multipliers Idea: Recall that for a continuous function y = f(x) on a closed and bounded interval a x b, we optimize f(x) with the following two facts: (a) f(x) must attain it minimum and maximum for some values of x in the interval a x b. (b) The minimum and maximum of f(x) occurs at the end points (i) x = a, b or at (ii) critical points in a < x < b. The range a x b of values of x is the constraint on which f(x) is optimized. However for multivariable functions the constraint may be some complicated relation satisfied by the indecent variables. For example, in the hiking exercise you did you are reading the highest point on your path above sea level and the lowest point on your path below sea level. In that context, the constraint is the hiking path on the xy plane while the function you are optimizing is the height function. y z = 20 C A B z = 40 z = 50 z = 30 z = 10 z = 0 E D z = 0 z = -5 z = -20 z = -10 G F 0 x Let the height function be given by z = f(x, y) and the equation of the (projected) path be g(x, y) = 0. From geometric considerations, we see that the constraint curve and the contour curve at a possible min or max must share the same. Therefore the critical points on the path are given by the equations: g (x, y) = λ (x, y) (1) x x (x, y) = λ g (x, y) (2) g(x, y) = 0 (3) where x, y, and λ are to be determined. Here (x, y) are the critical points. Note that Equation (3) ensures the solution is on the constraint curve. The first two equations are called Lagrange Multipliers. 4

5 If the constraint path is closed and bounded (either a closed loop or curve including end points without self crossings) then the function f(x, y) must attain minimum and maximum at some points on the path. 2a. The height of the slanted roof a house is given h(x, y) = 2x + 4y A spider on the roof is observed from the top view crawling on the closed path x 2 + y 2 = 4. What is the minimum and maximum height attained by the spider? In this context, the function we need to optimize is the height h(x, y) = 2x + 4y + 20 with constraint x 2 + y 2 4 = 0. Can you roughly draw a picture to depict the path of the spider on the roof showing where the graph of x 2 + y 2 = 4 is in relation to the actual path of the spider? Use Lagrange multipliers to find the minimum and maximum heights of the spider. 2b. How would you change your answer if the spider only crawled on the path that tracks the upper semi-circular part of the curve x 2 + y 2 = 4? 5