Determine the following quantities fxx y fyx y and fxyx y

Question

1. Limits and Continuity:

a. Sketch the contour plot for the following functions, and then show that the limit as (x, y) → (0,0) does not exist by determining at least two paths of approach which have different limits as (0. 0):

i. f(x,y) = y/(1-e^x)

ii. f(x,y) = 2x²y / (x⁴+y²)

iii. f(x,y) = x²+y² / y

2. Partial Dervatives:

a. Consider the following multivariate function:

f(x, y) = 2xy + √(x+4y).

i. Determine the following quantities: f_x(x, y), f_y{x. y). and f_xy(x, y), f_xy(x, y) and f_yy(x, y).

ii. Determine the equation of the tangent plane at the point (-2. 1). Use it to approximate the value of f(-1. 99, 1. 01).

b. Consider the following multivariate function:

f(x,y) = (x² - y²) / (x+y)

i. Determine the following quantities: f_x(x, y), f_y{x. y). and f_xy(x, y), f_xy(x, y) and f_yy(x, y).

ii. Determine the equation of the tangent plane at the point (0, 1). Use it to approximate the value of f(0. 1, 0. 9).