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Additional exercises 2

Exercise

Let f(x)=(x,x2,x3) and g(x,y,z)=x+yz. Identify the domain and codomain of f, g. Determine if fg and gf are well defined. If it is well defined then write the function explicitly and determine the domain and codomain.

Exercise

Let f(x,y)=x2+5xy+lny. Compute the partial derivatives fx, fy.

Exercise

Let f(x,y)=x3+5+exy. Calculate f, the gradient of f.

Exercise

Determine the directional derivative, Dvf(1,4,6) for f(x,y,z)=exy2+4zy3 in the direction of v=(2,3,6).

Exercise

Compute the Jacobian matrix of the transformation

(u,v)(eucosv,eusinv).

Exercise

Consider the functions,

α:[0,)R2;t(tcost,tsint),f:R2R;(x,y)3x2+y2.

Let g=fα. Calculate g both by using the chain rule and by first calculating g and then differentiating and confirm that the answer is the same using either method. Out of curiosity, try to understand what curve α(t) traces out as t varies.

Exercise

Let f be defined, for (x,y)(0,0), as

f(x,y)=xyx2+y2,

and f(x,y)=0. Calculate the 4 missing values in the following table, assuming that t0.

linevalue
{x=0}f(0,t)= ?
{y=0}f(t,0)= ?
{x=y}f(t,t)= ?
{x=y}f(t,t)= ?

What does this say about f when approaching (0,0) along these different lines?

Exercise

Consider the surface x2+y2z2=1. Verify that the point (1,1,1) is contained in the surface. Find the tangent plane to this surface at this point.

Hint: write this surface as a level set {(x,y,z):f(x,y,z)=c}, calculate f at the specified point and use the known connection between gradient and tangent plane.