Multivariate Calculus

Problem 1

If f(x,y,z)=(x^2)(y^2)-((z^3)/y), find the directional derivative of f at (-1,1,5) in the direction of <3,1,2>.

Problem 2

Find the normal line and tangent plane to z=2 x y+y cos(x (y^3)) at the point ((π/2),1,π)

Problem 3

Find the normal line and tangent plane to the surface (x^2)y+(Sqrt[1+y z])+x z=6 at the point (1,0,5).

Problem 4

Use the cross product to find the equation of the plane through (1,2,1), (3,1,4), and (2,4,0)

Problem 5

Find and classify the potential extreme values of f(x,y) =6 x y-2 (x^2)y-x (y^2)

Problem 6

Show that for x>0 and y>0, the function f(x,y)=(x/y)-x y has no local maxima or minima.

Problem 7

Find  ((Underscript[∫∫, R])^( ))sin(x+3y) dA, where R is the rectangle 0≤x≤π and 0≤y≤(π/2).

Problem 8

Find  ((Underscript[∫∫, R])^( ))x (y^2)+4y dA, where R is the region bounded by y=2x, y=0, and x=6.  Do the integral in both orders of integration.

Problem 9

Find (∫_C)(2x-y)dx+(3x-2y)dy, where C is the curve parameterized by x=(e^t), y=(t^2), as t goes from 0 to 1.

Problem 10

Find (∫_C)((x^2)+3y)dx+((x^2)-10y)dy, where C is the counterclockwise-oriented boundary of the square from (-1,-1) to (1,1).

Problem 11

Find  (∫_C)((x^3)+6x y+(y^2))dx+(3(x^2)+2x y-5)dy, where C is the ellipse ((x^2)/9)+((y^2)/16)=1 parametrized counterclockwise.

Multivariate Calculus Concepts

Partial derivatives

Directional derivatives

Gradient of a function

Equations of lines and planes

Tangent planes and normal lines to a surfaces

Finding and classifying local extrema of a function of 2 variables

Multivariate integrals

Line integrals

Green’s Theorem