MAT 21C Lecture Notes - Lecture 12: Binomial Series, Cartesian Coordinate System, Ellipse
MAT 21C – Lecture 12 – Binomial Expansion and Introduction to Vectors
• Binomial Expansion: Let f(x) = (1 + x)p ,f’x = p1 + xp-1, f’’x = pp – 1)(1 + x)p-2,
fn(x) = p(p – 1)(p – 2)…p – n + 1)(1 + x)p-n. Thus,
•
where c is between 0 and x. as
when The binomial series converges to .
• Example: When p = 2, we have . All other terms are 0.
When p = -1,
This is a geometric series with ratio, r = -x.
When p = ½, then
• Consider
Then
for some c between 0 and x. Thus,
• Example: Approximate
Applying the Taylor series for binomials with p = 1/2, we obtain
for some c between 0 and x. Then,
.
for
. In fact,
• To find error in Taylor Polynomial: . A Taylor polynomial
centered at a is
. Then
for some c
between a and x (a < c < x or x < c < a)
• Vectors will be introduced with coordinate systems in three dimensions and
Cartesian coordinates (x, y, z). A right-handed coordinate system is the default.
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Document Summary
Mat 21c lecture 12 binomial expansion and introduction to vectors. =(cid:2868: binomial expansion: let f(x) = (1 + x)p ,f"(cid:894)x(cid:895) = p(cid:894)1 + x(cid:895)p-1, f""(cid:894)x(cid:895) = p(cid:894)p 1)(1 + x)p-2, fn(x) = p(p 1)(p 2) (cid:894)p n + 1)(1 + x)p-n. Thus, (cid:4666)(cid:883)+(cid:1876)(cid:4667)= (cid:1876)=(cid:883)+ (cid:4666)(cid:2868)(cid:4667)! (cid:1876)+(cid:2869)(cid:2870)!(cid:4666) (cid:883)(cid:4667)(cid:1876)(cid:2870)+(cid:2869)(cid:2871)! (cid:4666) (cid:883)(cid:4667)(cid:4666) (cid:884)(cid:4667)(cid:1876)(cid:2871)+ (cid:4666)(cid:883)+(cid:1855)(cid:4667) (cid:2869) where c is between 0 and x. (cid:4666)(cid:1876)(cid:4667) (cid:882) as: (cid:4666)(cid:1876)(cid:4667)= (cid:4666) (cid:2869)(cid:4667) (cid:4666) (cid:4667) (cid:4666)+(cid:2869)(cid:4667)! The binomial series converges to (cid:4666)(cid:883)+(cid:1876)(cid:4667) if |(cid:1876)|
