9.2: Taylor Series
Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series.
Brook Taylor 1685 - 1731
Greg Kelly, Hanford High School, Ri
7.4 Lengths of Curves and Surface Area
(Photo not taken by Vickie Kelly)
Greg Kelly, Hanford High School, Richland, Washington
Lengths of Curves:
ds dx
dy
If we want to approximate the length of a curve, over a short distance we could measure a
9.2 day 2 Maclaurin Series
Liberty Bell, Philadelphia, PA
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
There are some Maclaurin series that occur often enough that they should be memorized. They are on your form
7.5 part 1 Work and Pumping Liquids
Greg Kelly, Hanford High School, Richland, Washington
Review: Hooke's Law:
F
kx
A spring has a natural length of 1 m. A force of 24 N stretches the spring to 1.8 m.
a Find k:
F kx 24 k .8 30 k F 30 x
b How m
8.1: L'Hpital's Rule
Actually, L'Hpital's Rule was developed by his teacher Johann Bernoulli. De l'Hpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons.
Guillaume De l'Hpital 1661 - 1704
Greg K
7.3 day 2 Disks, Washers and Shells
Limerick Nuclear Generating Station, Pottstown, Pennsylvania
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
2
y
1
x
Suppose I start with this curve. My boss at the ACME Rocke
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sX 6 Ypg 6d6 eF 6 ~ 674 ` 6 v5FD7U2PlaU1#00g c U`Uf x74 ` #P5 b z f` f R T r V T S RQ I H G X ` W G p W W c 6 W 6 p 6 z f sX 6 Yg bd6 {`F 6 ~ 74 ` 6 f7%0D7U2PlaU1#00g c U`Uf x74 ` #P5 T R r V T S RQ I H G X ` W G p W W c 6 W
u~lw k c!Ewe R p P a R y d H w H f a a p f d r P p a dW a a s Hh r R Y f y H Rw t H R R a rW s P R Y R f EXzly!wb"uX!CEX!S`EXu"%Eu~vwUc8yQv`I"ulEX`wVcV8 r fW a HW a r R P R W T h f P R T P f R a r f T r R R T d R W a H W P R
13.1 Vector-valued function - A vector valued function is a function whose domain is all real numbers but its range is a set of vectors. r(t) = <f(t) , g(t), h(t)> r(t) = f(t)i-> + g(t) j-> + h(t) k -> r(t) = <1+t, 3t, -t> r(2) = <3,6,-2> ^ ^
real nu
13.4 Motion in 3-D - Path of the moving object r->(t) = <2-t,4(t)1/2> Suppose @ P(t=1) <= The equation of the path *t is time
- Velocity of the object? - Acceleration of the object? - Speed? r->(t) = r-> ' (t) = <-1, 2t-1/2> = <-1, 2/(t)1/2> |t=1 =
Greg Kelly, Hanford High School, Richland, Washington
A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below.
What is the total distance traveled by the bee?
200 200 200 100 700
100
700 feet
ft min 50
7.2 Areas in the Plane
Gateway Arch, St. Louis, Missouri
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
y1
2 x
2
How can we find the area between these two curves?
y2
x
We could split the area into several s
8.3 day one
Improper Integrals
Greg Kelly, Hanford High School, Richland, Washington
Until now we have been finding integrals of continuous functions over closed intervals.
Sometimes we can find integrals for functions where the function or the li
8.4 day 2
Trigonometric Substitutions
Greg Kelly, Hanford High School, Richland, Washington
We can use right triangles and the pythagorean
theorem to simplify some problems.
1
∫
dx
4+ x
2
These are
in the same
form.
4 x
a + x
ln secθ + tan θ + C
4+ x
x
+
8.3 day 2 Tests for Convergence
Greg Kelly, Hanford High School, Richland, Washington
Review:
1
dx P x
P 0
(P is a constant.)
b lim b P 1
P 1
P 1
1 P 1
P 1
1
x
b 1
P
dx
P
If P 1 then b
gets bigger and bigger as b , therefore the integr
8.2 Relative Rates of Growth
Greg Kelly, Hanford High School, Richland, Washington
The function y
e x grows very fast.
We could graph it on the chalkboard:
If x is 3 inches, y is about 20 inches:
35 30 25 20 15 10 5 0
3, 20
We64 inches, the y-
9.1 Power Series
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
Start with a square one unit by one unit:
1 2
1 4
1 1 1 1 8 16 32 64
1 8
1
1 32
1 64
1 2 1 4
1
1 16
1
This is an example of an infinite se
8.4 Partial Fractions
The Empire Builder, 1957
Greg Kelly, Hanford High School, Richland, Washington
1
5x 3 dx 2 x 2x 3
5x 3 x 3 x 1
This would be a lot easier if we could re-write it as two separate terms.
A x 3
B x 1
Multiply by the common
7.5 Fluid Pressure and Forces
Greg Kelly, Hanford High School, Richland, Washington
What is the force on the bottom of the aquarium?
2 ft
1 ft
3 ft
Force weight of water density volume lb 62.5 3 2 ft 3 ft 1 ft ft
375 lb
If we had a 1 ft x 3 ft
9.5
Testing Convergence at Endpoints
The original Hanford High School, Hanford, Washington
Greg Kelly, Hanford High School, Richland, Washington
Remember:
The Ratio Test:
If
an is a series with positive terms and lim
n
an 1 an
L
then:
The s
9.4
Radius of Convergence
Greg Kelly, Hanford High School, Richland, Washington
Convergence The series that are of the most interest to us are those that converge.
Today we will consider the question:
"Does this series converge, and if so, for wh
9.3 Taylor's Theorem: Error Analysis for Series
Tacoma Narrows Bridge: November 7, 1940
Greg Kelly, Hanford High School, Richland, Washington
Taylor series are used to estimate the value of functions (at least theoretically - now days we can usual
7.3 Day One: Volumes by Slicing
Little Rock Central High School, Little Rock, Arkansas
Photo by Vickie Kelly, 2001 Greg Kelly, Hanford High School, Richland, Washington
Find the volume of the pyramid: 3 Consider a horizontal slice through the pyram
Chapter 7 Extra Topics
Crater Lake, Oregon
Photo by Vickie Kelly, 1998 Greg Kelly, Hanford High School, Richland, Washington
Fg
Centers of Mass:
d
Torque is a function of force and distance. (Torque is the tendency of a system to rotate about a p
14.4 Tangent Plane & Linear approximation ex: z = f(x.y) = ln(x-3y) @ P(7.2) = P(7,2,0) Evaluate f(6.9, 2.06) - you need to approximate Approximate f(6.9,2.06) = ~ - 0.1 EQ of the tangent plane to z = ln (x - 3y) @ (7,2) z - z0 = fx(x0,y0) (x-x0) + f