## Parametric EquationsEdit

*P*(0,0) to

*Q*(7,17).

*x*-axis and the minor axis of length 3 along the

*y*-axis, generated clockwise.

## Polar CoordinatesEdit

*y*=

*mx*+

*b*in polar coordinates.

Sketch the following polar curves without using a computer.

Sketch the following sets of points.

## Calculus in Polar CoordinatesEdit

Find points where the following curves have vertical or horizontal tangents.

Sketch the region and find its area.

## Vectors and Dot ProductEdit

## Cross ProductEdit

Find and

Find the area of the parallelogram with sides and .

*x*-axis. Find the magnitude and the direction of the torque at the pivot when the force is applied to the wrench

*n*units away from the origin.

Prove the following identities or show them false by giving a counterexample.

## Calculus of Vector-Valued FunctionsEdit

## Motion in SpaceEdit

## Length of CurvesEdit

Find the length of the following curves.

## Parametrization and Normal VectorsEdit

**T**and the principal unit normal vector

**N**for the curve Check that

**T**⋅

**N**=0.

## Equations of Lines And PlanesEdit

*x*−

*y*+

*z*=1 passing through the point (0,2,-2)

*x*+

*y*+2

*z*=4 passing through the point (5,5,5).

*x*+2

*y*−

*z*=1 and

*x*+

*y*+

*z*=1 intersect.

*x*+2

*y*−

*z*=1 and

*x*+

*y*+

*z*=1.

*x*+

*y*+

*z*=1.

## Limits And ContinuityEdit

Evaluate the following limits.

At what points is the function *f* continuous?

Use the two-path test to show that the following limits do not exist. (A path does not have to be a straight line.)

## Partial DerivativesEdit

Find the four second partial derivatives of the following functions.

## Chain RuleEdit

Find

Find

*x*is the side of the square base and

*h*is the height of the pyramid. Suppose that and for Find

## Tangent PlanesEdit

Find an equation of a plane tangent to the given surface at the given point(s).

## Maximum And Minimum ProblemsEdit

Find critical points of the function *f*. When possible, determine whether each critical point corresponds to a local maximum, a local minimum, or a saddle point.

Find absolute maximum and minimum values of the function *f* on the set *R*.

*R*is a closed triangle with vertices (0,0), (2,0), and (0,2).

*x*−

*y*+

*z*=2 closest to the point (1,1,1).

## Double Integrals over Rectangular RegionsEdit

Evaluate the given integral over the region *R*.

Evaluate the given iterated integrals.

## Double Integrals over General RegionsEdit

Evaluate the following integrals.

*R*is bounded by

*x*=0,

*y*=2

*x*+1, and

*y*=5−2

*x*.

*R*is in the first quadrant and bounded by

*x*=0, and

Use double integrals to compute the volume of the given region.

## Double Integrals in Polar CoordinatesEdit

*R*is the unit disk centered at the origin.

## Triple IntegralsEdit

In the following exercises, sketching the region of integration may be helpful.

*x*+3

*y*+6

*z*=12 and the coordinate planes.

*y*=

*x*and

*x*=0.

*dydzdx*.

## Cylindrical And Spherical CoordinatesEdit

*z*=0 and the hyperboloid

*D*is a unit ball, use a triple integral in spherical coordinates to evaluate

## Center of Mass and CentroidEdit

## Vector FieldsEdit

One can sketch two-dimensional vector fields by plotting vector values, flow curves, and/or equipotential curves.

## Line IntegralsEdit

*C*is the line segment from (0,0) to (5,5)

*C*is the circle of radius 4 centered at the origin

*C*is the helix

*C*is the arc of the parabola

## Conservative Vector FieldsEdit

Determine if the following vector fields are conservative on

Determine if the following vector fields are conservative on their respective domains in When possible, find the potential function.

## Green's TheoremEdit

*y*=0 and below

*y*=

*x*(2-

*x*) in two different ways, and compare the answers.

## Divergence And CurlEdit

## Surface IntegralsEdit

*z*=2−

*x*−

*y*in the first octant.

*z*direction.

*y*direction.

## Stokes' TheoremEdit

where , is the upper half of the ellipsoid , and points in the direction of the

*z*-axis.

where , is the part of the sphere for , and points in the direction of the

*z*-axis.

## Divergence TheoremEdit

Compute the net outward flux of the given field across the given surface.

*xy*-plane.