Calculus/Multivariable and differential calculus:Exercises

Polar Coordinates

Sketch the following polar curves without using a computer. Template:Question-answer Template:Question-answer Template:Question-answer

Sketch the following sets of points. Template:Question-answer Template:Question-answer

Calculus in Polar Coordinates

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

Sketch the region and find its area.

Cross Product

Find $𝐮\times 𝐯$ and $𝐯\times 𝐮$

Find the area of the parallelogram with sides $𝐮$ and $𝐯$.

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

Length of Curves

Find the length of the following curves.

Limits And Continuity

Evaluate the following limits. Template:Question-answer

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 Derivatives

Find the four second partial derivatives of the following functions.

Chain Rule

Find $df/dt.$

Find $f_s,f_t.$

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Tangent Planes

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

Maximum And Minimum Problems

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. Template:Hidden

Find absolute maximum and minimum values of the function f on the set R. Template:Hidden

Double Integrals over Rectangular Regions

Evaluate the given integral over the region R.

Evaluate the given iterated integrals.

Double Integrals over General Regions

Evaluate the following integrals.

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

Triple Integrals

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

Vector Fields

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

Conservative Vector Fields

Determine if the following vector fields are conservative on ${ℝ}^2.$

Determine if the following vector fields are conservative on their respective domains in ${ℝ}^3.$ When possible, find the potential function.

Divergence Theorem

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

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