Concept
Equations/Formulas
r(t ) r0 tv
Vector Equation of a Line
x xo tv1
Parametric Equation of a Line
y y o tv 2
z z o tv 3
A( x xo ) B( y yo ) C ( z z o ) 0
Equation for a Plane in Space
d
Distance from a Point to a Line in Space
Line of Intersection o
Functions of Several Variables (14.1)
Fully Worked Examples
Problem: Determine the domain of
Solution:
We cannot take the logarithm of a negative number or 0, therefore,
Problem: Identify the level curves of
Solution:
This is the equation of a circle of r
Chain Rule (14.4)
Fully Worked Examples
Problem: Compute
for
Solution:
Define Rule
Perform derivative
Problem: Find
if
,
Solution:
Note this is a helix (See Section 13.1)
Define rule
Perform derivative
,
,
Problem: Find
in terms of r and s if
Solutio
Partial Derivatives (14.3)
Fully Worked Examples
Problem: Find
and
for
. Also find the slope of the
traces at the point
Solution:
Find
Evaluate at point
Find tangent line for traces.
The tangent line at
slope of
.
for the trace of
for the plane
has a
T
Curvature (13.4)
Fully Worked Examples
Problem: Using the previous examples from 13.3, find the unit tangent vector for
r(t).
Solution:
Find the parameterization of r in terms of s
Find
Check if
is same as
Problem: Find T for the vector
Solution:
Find
Components of Acceleration (13.5)
Fully Worked Examples
Problem: Find the acceleration of motion (without T or N) given the position
function
.
Solution:
Find
Find
Find a
Problem: Find the B vector and the torsion for the position function
.
Solution:
Limits and Continuity (14.2)
Fully Worked Examples
Problem:
Solution:
Function is undefined at points
Use standard substitution
since we will get division by zero.
Problem:
Solution:
Function value does not exist at the point in question, therefore sub
Area by Double Integral (15.3)
Fully Worked Examples
Problem: Find the area of the region
quadrant.
bounded by
Solution:
Graph region and find where curves intersect
Set up integral
Integrate
and
in the first
Problem: Find the area of the region
line
e
Moments and Center of Mass (15.6)
Fully Worked Examples
Problem: Find the center of mass of a solid of constant density bounded below
by the disk
in the plane
and above the paraboloid
Solution:
Sketch graph
Note: constant density means
Find
Change to p
Double Integrals in Polar Form (15.4)
Fully Worked Examples
Problem: Change the integral
into an equivalent polar
integral
Solution:
Look at limits on graph and change limits to polar
- Note: The limits on the x go from 0 to 1 representing only half of t
Triple Integrals of Rectangular Coordinates (15.5)
Fully Worked Examples
Problem: Find the volume of the region D enclosed by the surfaces
and
Solution:
Sketch graph
Find shadow of 3D object in xy plane and call this region R
Find z-limits by drawing l
Integration and Projectile Motion (13.2)
Fully Worked Examples
Problem: Find
Solution:
-Problem: Find
Solution:
Problem: What is the position vector of a hang glider that has an acceleration
vector of
At time
the glider departed
from the point (3,0,0) wit
Cylinders and Quadratic Surfaces (12.6)
Problem: Identify and sketch the surface
Solution:
Put equation in standard form (move 4 to right and divide everything by -4)
Look at curves for each coordinate axis
o
:
o
:
- Hyperbola in yz plane
Hyperbola in
Calculus of Parametric Equations (11.2)
Fully Worked Examples
Problem: Find the tangent line(s) to the parametric curve given by
at the point
Solution:
Find
Determine the values of t that will give the point given.
Thus,
tangent line)
(ts must appear in
Parametric Equations (11.1)
Fully Worked Examples
Problem: Find a Cartesian equation from the following set of parametric
equations:
Solution:
Solve for t with one of the equations
Substitute into the other equation and solve for x or y (whichever is
ea
Lines and Planes, Pt. 1 (12.5)
Fully Worked Examples
Problem: Find the equation of the line that passes through the points
and
. Also write down the parametric form of the equation of the line.
Solution:
Find
Plug into vector equation formula
Parametri
Cross Product (12.4)
Fully Worked Examples
Problem: Find a
Solution:
Note: b
if
and
Problem: Find the area of the triangle with vertices
.
Solution:
Sketch problem
Find vector
and
Find cross product of
Find
and
and
Triangle is of a parallelogram so
P
Directional Derivative and Gradients (14.5)
Fully Worked Examples
Problem: Find the directional derivative for
the direction of
Solution:
Find unit vector of
Find dot product
in
Problem: Find the directional derivative,
in the direction of
Solution:
Fi