Question

Problem 1

WITRUUNIMLJ. Muues 14 and 15 Geometric applications and some 3D plotting You should strategically employ Symbolab/Wolfram for evaluating and checking symbolic things. Octave for ALL 3D plotting, please - this is one of our "demonstrate you can use Octave (MATLAB)" | assignments. Don't forget to include both scripts and plots. Problem 1: Consider the function f(x, y) = 2x? In(y)-3x*y at the point (a,b,f(a,b)) = (1,1, S (1,1)). (a) Find f(x,y), S.(a,b), S (x,y), ,(a,b), and f(a,b). (b) Give a vector tangent to the surface at the point, and in the direction of the y axis. Then, write the equation of the tangent line that has that vector as its direction vector. (c) Give a vector tangent to the surface at the point, and in the direction of the x axis. Then, write the equation of the tangent line that has that vector as its direction vector. (d) Give a vector normal to the surface at the point. Then, write the equation of the tangent plane that has that vector as its normal. (e) Produce an Octave plot of the surface, the two tangent lines, and the tangent plane. Recommend setting up mesh from 0 to 2 for both x and y. That will center the plot around the point of interest at (1,1), and avoid the regrettable side effects of log of negative number."

Answer 1

1)

a) From the function, we get

$\\ f_x(x,y) = \frac{\partial (2x^2 \ln y - 3x^2y^2)}{\partial x} = 4x \ln y - 6xy^2 \\ f_y(x,y) = \frac{\partial (2x^2 \ln y - 3x^2y^2)}{\partial y} = 4\cdot \frac{x^2}{y} - 6yx^2$

Hence,

$\\ f_x(a,b) = 4a \ln b - 6ab^2 = -6 \\ f_y(x,y) = 4\cdot \frac{a^2}{b} - 6ba^2 = -2 \\ f(a,b) = 2a^2\ln b - 3a^2b^2 = -3$