Calculus III Exam Review
Ashish Heda
Peter Zhu
List of Topics
11.5
• Equation of Line
• Equation of Plane
• Types of Lines/Planes
• Distance (from point to line/plane or from
line to plane)
• Line/Angle of Intersection between Planes
11.6
• Visualize/Name the Equation
12.1
• Rules for Differentiating Vector Functions
• Velocity. Speed, Acceleration
12.2
• Integration of Vector Functions
• Initial Value Problem/Projectile Motion
• Rules for Integrating Vector Functions
12.3
• Arc Length
• Arc Length Parameterization
• Speed On Smooth Curve
12.4
• Unit Tangent Vector
• Normal Vector
• Curvature
• Osculating Circle & Radius of Curvature
12.5
• Tangential and Normal Acceleration
• Binormal Vector
• Torsion
12.6
• Velocity and Acceleration in Polar
Coordinates
Possible Tricky Problems
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Equation of Line
Recall that in 2D the equation of a line is given by:
where m represents the slope
In three dimensions we have a similar equation:
*Note the bolded terms show these are vectors rather
than scalar values
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Equation of Plane
You need a way of describing every point on a
plane. What is something every line on the
plane will have in common?
Answer: all lines make a 90° angle with the
normal vector of the plane and thus, the dot
product of the normal line and any line on the
plane is 0.
Continued
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Types of Lines
Types of Planes
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Distance From Point to Line
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Distance From Point to Plane
Similar to finding distance from point to a line,
except that you are given the slope of the
perpendicular line!
How can you find a Point P on
the plane?
Use the equation of the plane and set any two variables to 0 and
solve for the other:
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Line/Angle of Intersection of Two
Planes
In order to determine the angle/line of intersection
take a look at the following picture:
The normal vectors of two planes
can be easily obtained.
What can you say about the angles
depicted in the picture to the left?
How can you determine the angle between the two
normal lines?
How can you determine the slope of the line of
intersection?
Now all you need is a point on the line and
you can find the line of intersection.
From here you can find theta and determine the angle
between the planes. Remember that there are
always 2 possible angles between planes! Usually
you are asked to provide the acute angle.
Line of Intersection
Now we need a point. This is any point that satisfies
both of the above equations. There are lots of
methods for doing this but here is one:
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Visualize/Name the Equation
What type of equation is it? What does it look
like?
Parabola that
goes forever
along the x-axis
Cylinder (with elliptical base)
Ellipsoid
Paraboloid
Hyperboloid (two
unconnected regions)
Hyperboloid (two
connected regions)
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Rules for Differentiation of Vector
Functions
Velocity, Speed, Acceleration
Problem 23 from section 12.1
• Does the particle have constant speed? If so, what is it?
• Is the particle’s acceleration vector always orthogonal to
its velocity vector?
• Does the particle move clockwise or counterclockwise
around the circle?
• Does the particle begin at the point (1,0)?
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Integration
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Initial Value Problems of Vector
Functions
Projectile Motion
A projectile is fired with an initial speed of 500 m/s at an
angle of elevation of 45°
• When and how far away will the projectile strike?
• How high overhead will the projectile be when it is 5 km
downrange?
• What is the greatest height reached by the projectile?
First, thing is convert this into a problem like the one before:
Possible information you may need to
know
• Acceleration due to gravity in m/s2 and ft/s2
• How many meters in a kilometer (km) …
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Rules for Integration of Vector
Functions
(see problem 33 of 12.2 for more material)
Note that there is no property listed for a nonconstant vector crossed with r(t).
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Arc Length Along a Curve
Recall that for 2D, the arc length is given by:
Thus, it makes sense that in 3D, the arc length is given by:
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Arc Length Parameterization with base
point
Sometimes we want to know what is the arc
length from an initial position to any given
position at time, t.
Ex. Tracking a satellite: you know its initial
position where it was put into outer space and
you know the path it will follow. You want to
design a formula so that at any given time you
know how much distance it has traveled
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Find the arc length parameter along the curve
form the point where t = 0 for the curve:
Then, re-write the curve as function of the arc
length parameter.
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Speed on Smooth Curve
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Unit Tangent Vector
Represents the change in
direction of the curve with
respect to time or the
direction of the velocity at
anytime, t.
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Normal Vector
This is change of the change in
direction of the curve with
respect to time or the
change in of the direction of
velocity with respect to
time.
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Binormal Vector
Has physics and other
applications but it is
important for completing the
Frenet frame. This is similar to
the xyz frame except unlike
the xyz frame this frame
travels with the curve.
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Cuvature
The rate at which the direction of the curve
changes with respect to the distance traveled.
Best way to think about curvature is driving a
car. If the direction of travel changes from
driving north to driving east in 1 mile then it
not that difficult of a turn to make. On the
other hand going from north to east in a
matter of 100ft is much harder and thus a
much greater curvature.
You can also think of it in terms of driving along a
circle: Imagine you are going at 100 mph around
the following two circles:
R=100 ft
R=5,280ft
Harder to drive around
Easier to drive around
Thus, curvature can be approximated by:
The actual formula is:
Computational Formula of Curvature
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Osculating Circle or Circle of Curvature
and the Radius of Curvature
• Tangent to the curve at P
(has the same tangent line
the curve has)
• Has the same curvature
the curve has at P
• Has center located along
the direction of the
Normal vector.
• Radius of curvature =
Tangential and Normal Components of
Acceleration
An easy way to understand this is that the
tangential component of a(t) points in the
direction of the unit tangent vector (or the
velocity vector) and similarly the normal
component of a(t) points in the direction of
the normal vector.
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Torsion
Rate at which the osculating plane turns about T
as P moves along the curve. Measures how
the curve twists.
Computational formula
Possible Tricky Questions
(True/False)Three statements:
I. Two perpendicular lines to a plane are parallel
II.
III. The derivative of the arc length is the velocity.
(answer: T, T, F (derivative of arc length is speed)
(Types of Planes, Rules of differentiation or
integration, acceleration components, arc
length, and curvature concepts…)
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• May give you 4 points and ask you if all lie on
the same plane
• Distance from a line to a plane (be careful to
check that the line does not intersect the plane)
• Ask you to re-write a curve with the arc length
parameter (this means you should provide r(s))