Vector Calculus for Engineers CME100, Fall 2004 Handout #2 Functions of Several Variables, Partial Derivatives, Chain Rule 1. Find the limit: lim x , y 0 xy by examining the following x y2 2 cases: a) approaching (0,0) along y 0 b) approaching (0,0) along x 0 c) approaching (0,0) along y mx 2. For f ( x, y, z) xy2 sin( yz) find f x , f y , f xx , f yy , f zz , f xy and f yx . >> syms x y z >> diff('x*y^3*sin(y*z)',x) y^3*sin(y*z) >> diff('x*y^3*sin(y*z)',y) 3*x*y^2*sin(y*z)+x*y^3*cos(y*z)*z >> diff('x*y^3*sin(y*z)',z) x*y^4*cos(y*z) f f and in r polar coordinates x r cos and y r sin . 3. For the function f ( x, y ) determine 4. For w( x, y, z ) xy z such that x y z determine w w a) b) x y x z Linearization, Differential 5. Two resistors R1 and R2 are connected in parallel. If R1 200 and R2 300 and if these values may be in error by as much as 2 , estimate the maximum error in the calculated value of the combined resistance. Gradient, Directional Derivative 6. For the following function z f ( x, y) 10 x 2 y 2 determine: a) gradient of z at (1,1) b) directional derivative in the direction of 3i 4 j u at (1,1) 5 c) directions of maximum increase, maximum decrease, and no change in z d) a particle moves by a distance of 0.1 units in the direction of the gradient. Determine the approximate change in z e) a vector normal to the surface z 10 x 2 y 2 at (1,1) meshc makes a surface plot with level curves for a function in 3D >> [x,y]=meshgrid(-2:0.1:2); >> z=10-x.^2-y.^2; >> meshc(x,y,z) Maxima and Minima, Lagrange Multipliers 7. Find the least amount of plywood needed to construct a closed rectangular box of a given volume V. z Function fminsearch minimizes functions of several variables x y >> fminsearch('2*x(1)*x(2)+2/x(1)+2/x(2)',[1,2]) ans = 1.0000 1.0000 8. Determine an equation of a line that best fits n data points ( x1 , y1 )...( xn , y n ) in the least squares sense. Solve for the slope and the intercept for the following set of data points: 3 2.5 2 1.5 1 0.5 n 1 2 3 4 x 0 1 2 3 y 0 2 1 2 Function polyfit determines best fit (linear and nonlinear) to a given set of data >> x=[0 1 2 3]; >> y=[0 2 1 2]; >> polyfit(x,y,1) ans = 0.5000 0.5000 0 -0.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 9. The temperature over a semicircular disk of radius 1 is given by: T ( x, y) xy x 2 . Find all absolute maxima and minima (if any) over the specified region. 10. Redo Problem 7 using the method of Lagrange Multipliers.