Discuss teh below:
Q: If I have the following function f(x,y) = x^3 + x^2y - 16y
There are 6 things I would like to know how to do:
How to find the ?rst-order and second-order partial derivatives of f?
If the value of f when x = 3 and y = -2 is 41 but I was to know only that the value of x lies between 2.98 and 3.02, while the value of y lies between -2.03 and -1.97, how would I use the ?rst-order partial derivatives to investigate the accuracy of 41 as an estimate of f(x,y) for values of x and y in these ranges?
How would I use the chain rule to determine, in terms of t, the rate of change of f with time along the path given at time t by x(t) = 4 sec(t), y(t) = 1 - t^2?
If I have a surface where surface z = f(x,y) where f(x,y) is de?ned above how would I determine the gradient function grad f(x,y) and hence ?nd the slope of the surface at the point (3, -2,41) in the direction of the unit vector j?
How would I determine the second-order Taylor polynomial for f about (-1,2)?
And finally how would I locate and classify all the stationary points of f?