Calculus/Multivariable and differential calculus:Exercises
- 1 Parametric Equations
- 2 Polar Coordinates
- 3 Calculus in Polar Coordinates
- 4 Vectors and Dot Product
- 5 Cross Product
- 6 Calculus of Vector-Valued Functions
- 7 Motion in Space
- 8 Length of Curves
- 9 Parametrization and Normal Vectors
- 10 Equations of Lines And Planes
- 11 Limits And Continuity
- 12 Partial Derivatives
- 13 Chain Rule
- 14 Tangent Planes
- 15 Maximum And Minimum Problems
- 16 Double Integrals over Rectangular Regions
- 17 Double Integrals over General Regions
- 18 Double Integrals in Polar Coordinates
- 19 Triple Integrals
- 20 Cylindrical And Spherical Coordinates
- 21 Center of Mass and Centroid
- 22 Vector Fields
- 23 Line Integrals
- 24 Conservative Vector Fields
- 25 Green's Theorem
- 26 Divergence And Curl
- 27 Surface Integrals
- 28 Stokes' Theorem
- 29 Divergence Theorem
Sketch the following polar curves without using a computer.
Sketch the following sets of points.
Calculus in Polar Coordinates
Find points where the following curves have vertical or horizontal tangents.
Sketch the region and find its area.
Vectors and Dot Product
Find the area of the parallelogram with sides and .
Prove the following identities or show them false by giving a counterexample.
Calculus of Vector-Valued Functions
Motion in Space
Length of Curves
Find the length of the following curves.
Parametrization and Normal Vectors
Equations of Lines And Planes
Limits And Continuity
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.)
Find the four second partial derivatives of the following functions.
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.
Find absolute maximum and minimum values of the function f on the set R.
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.
Double Integrals in Polar Coordinates
In the following exercises, sketching the region of integration may be helpful.
Cylindrical And Spherical Coordinates
Center of Mass and Centroid
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
Determine if the following vector fields are conservative on their respective domains in When possible, find the potential function.
Divergence And Curl
Compute the net outward flux of the given field across the given surface.