Non-degenerate real quadric surfaces
Ellipsoid
Spheroid
(special
case of
ellipsoid)
Sphere
(special
case of
spheroid)
Elliptic
paraboloid
Circular
paraboloid(
special
case of
elliptic
paraboloid)
Hyperbolic
paraboloid
Hyperboloi
d of one
sheet
Hyperboloi
INTEGRALS
TRIG INTEGRALS
RATIONAL FUNCTIONS
DERIVATIVES
BASIC RULES
o Sum/DifferenceRules
o ProductRule
o QuotientRule
TRIG
*
TRIG INVERSE(arccos = cos^-1)
Identities
Pythagorean
Double Angle
or
or
Sum And Difference
sin(A+B) = sin A cos B + cos A sin B
Formulas and Identities
Trig Cheat Sheet
Definition of the Trig Functions
Right triangle definition
For this definition we assume that
p
0 < q < or 0 < q < 90 .
2
Unit circle definition
For this definition q is any angle.
y
( x, y )
hypotenuse
y
opposite
Geometric Series- terms in the series are multiplied by the same
factor every time. It can be written in the form,
or with an index shift
Solution
1) First need to find a(the first term plug in n) and r(the rate
the series changes by)
If a and r are found
Formulas and Identities
Trig Cheat Sheet
Definition of the Trig Functions
Right triangle definition
For this definition we assume that
p
0 < q < or 0 < q < 90 .
2
Unit circle definition
For this definition q is any angle.
y
( x, y )
hypotenuse
y
opposite
Series Basics
Now back to series. We want to take a look at the limit of the
sequence of partial sums,
. Notationally well define,
We will call
an infinite series and note that the series starts
at
because that is where our original sequence,
, started.
H
INTEGRATION BY PARTS Is the integrand a polynomial times a trig function,
exponential, or logarithm? If so, then integration by parts may work.
Given
o Where f is a function that looks to have an easy derivative.
o Where g is a function that looks to be
Geometric Series- terms in the series are multiplied by the same
factor every time. It can be written in the form,
or with an index shift
Solution
1) First need to find a(the first term plug in n) and r(the rate
the series changes by)
If a and r are found
Geometric Series- terms in the series are multiplied by the same
factor every time. It can be written in the form,
(for series starting at n=1)
(for series starting at n=0)
Solution
1) First need to find a(the first term plug in n) and r(the rate
the seri
INTEGRALS
TRIG INTEGRALS
RATIONAL FUNCTIONS
DERIVATIVES
BASIC RULES
o Sum/DifferenceRules
o ProductRule
o QuotientRule
TRIG
TRIG INVERSE(arccos = cos^-1)
Identities
Pythagorean
Double Angle
or
or
Sum And Difference
sin(A+B) = sin A cos B + cos A sin B
cos
Identities
Pythagorean
Double Angle
or
or
Sum And Difference
sin(A+B) = sin A cos B + cos A sin B
cos(A+B) = cos A cos B sin A sin B
sin(AB) = sin A cos B cos A sin B
cos(AB) = cos A cos B + sin A sin B
Ellipsoid
Spheroid (special case of ellipsoid)
Sphere (special case of spheroid)
Hyperboloid of one sheet
Elliptic paraboloid
Circular paraboloid(special case of elliptic
paraboloid)
Hyperbolic paraboloid
Circular Cone (special case of cone)
Hyperboloid o
The Dot Product produces a scalar value(if dot =0 perp). The dot product of two vectors a=<a_1,a_2,a_3> and
b=<b_1,b_2,b_3> is given by
Projections One important use of dot products is in projections
.
The Cross Product is a vector product since it yields
INTEGRALS
TRIG INTEGRALS
RATIONAL FUNCTIONS
DERIVATIVES
BASIC RULES
o Sum/DifferenceRules
o ProductRule
o QuotientRule
TRIG
*
TRIG INVERSE(arccos = cos^-1)
Identities
Pythagorean
Double Angle
or
or
Sum And Difference
sin(A+B) = sin A cos B + cos A sin B
Distance from point to plane.
The distance from a point to a plane is equal to length of
the perpendicular lowered from a point on a plane.
If Ax + By + Cz + D = 0 is a plane equation, then distance
from point M(Mx, My, Mz) to plane can be found using the
Identities
Pythagorean
Double Angle
or
or
Sum And Difference
sin(A+B) = sin A cos B + cos A sin B
cos(A+B) = cos A cos B sin A sin B
sin(AB) = sin A cos B cos A sin B
cos(AB) = cos A cos B + sin A sin B
Geometric Series- terms in the series are multiplied by the same factor every
time. It can be written in the form,
(for series starting at n=1)
(for series starting at n=0)
Solution
1) First need to find a(the first term plug in n) and r(the rate the seri
INTEGRALS
TRIG INTEGRALS
RATIONAL FUNCTIONS
DERIVATIVES
BASIC RULES
o Sum/DifferenceRules
o ProductRule
o QuotientRule
TRIG
*
TRIG INVERSE(arccos = cos^-1)
Identities
Pythagorean
Double Angle
or
or
Sum And Difference
sin(A+B) = sin A cos B + cos A sin B
INTEGRALS
TRIG INTEGRALS
RATIONAL FUNCTIONS
DERIVATIVES
BASIC RULES
o Sum/DifferenceRules
o ProductRule
o QuotientRule
TRIG
TRIG INVERSE(arccos = cos^-1)
Identities
Pythagorean
Double Angle
or
or
Sum And Difference
sin(A+B) = sin A cos B + cos A sin B
cos
Geometric Series- terms in the series are multiplied by the same
factor every time. It can be written in the form,
(for series starting at n=1)
(for series starting at n=0)
Solution
1) First need to find a(the first term plug in n) and r(the rate
the seri
Geometric Series- terms in the series are multiplied by the same factor every
time. It can be written in the form,
(for series starting at n=1)
(for series starting at n=0)
Solution
1) First need to find a(the first term plug in n) and r(the rate the seri
Geometric Series- terms in the series are multiplied by the same
factor every time. It can be written in the form,
(for series starting at n=1)
(for series starting at n=0)
Solution
1) First need to find a(the first term plug in n) and r(the rate
the seri
TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 308H
Test 3 (take home, due 11:10 on Dec 1)
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work.
Name (print):
1.
Analyse the nonlinear system
x = x(3 x y ),
y = y (