Calculus-Basic Series Question?
If you can help on any of these, it would be so helpful! I'm stuck on these problems...maybe just a hint or if you do the problem out so I can learn, either would be appreciated! (except for telling me what series to use or something...that wouldn't really help.) 1.) Find the Taylor Polynomial of order 4 for f(x) = ln(1-x) at x=0 and use it to approximate f(.3) 2.) The Maclaurin series for f(x) is: 1+2x+((3x^2)/(2))+((4x^3)/(6))+..((n+1)x... a.)Find f " (0). b) Let g(x)= xf ' (x). Write the Maclaurin series for g(x) c.)Let h(x)= (the integral from 0 to x) f(t) dt. Write the Maclaurin series for h(x). 3.) The polynomial 1+7x+21x^2 is used to approximate f(x)=(1+x)^7 on the interval -0.01 ≤ x ≤ 0.01. a.) Use the Remainder Estimation Theorem to estimate the maximum absolute error. b.) Use a graphical method to find the actual maximum absolute error. 4.) Use Euler's Formula to write (i/2)(e^(3iθ)-e^(-3iθ) as a trigonometric function of θ. a.) sin3θ c.) cos3θ e.) 2sin3θ b.) 2cos3θ d.) -sin3θ Thank you so much!
Series Question Help! :)?
1) Let f(x) = e^x. The error is given by f^(5)(c) (0.28 - 0)^5/5! for some c in (0, 0.28). Since f^(5)(c) = e^c < e^(0.28), since f is an increasing function, the upper bound of the error is e^(0.28) * (0.28)^5 / 5!. --------------- 2) Note that this is the second order Maclaurin polynomial for f. So, the error is given by f '''(c) (x - 0)^3/3! for some c between 0 and x. However, f '''(c) = 8 * 7 * 6(1 + c)^5 = 336(1 + c)^5. So, |error| = |336(1 + c)^5 * x^3/3!| < 336 * (1 + 0.01)^5 * (0.01)^3 / 3!, since |x| ≤ 0.01 = 336 * (1.01)^5 * (0.01)^3 / 3!. ----------- 3) Use the geometric series. 1/(8+x) = 1/[8(1 + x/8)] ...........= (1/8) * 1/(1 - (-x/8)) ...........= (1/8) * Σ(n = 0 to ∞) (-x/8)^n ...........= (1/8) * Σ(n = 0 to ∞) (-1)^n x^n / 8^(n+1). ----------- 4) Σ(n = 1 to ∞) 8/(n(n+3)) = lim(t→∞) Σ(n = 1 to t) (8/3) [1/n - 1/(n+3)], by partial fractions = lim(t→∞) (8/3) [(1 - 1/4) + (1/2 - 1/5) + (1/3 - 1/6) + (1/4 - 1/7) + ... + (1/t - 1/(t+3))] = lim(t→∞) (8/3) (1 + 1/2 + 1/3 - 1/(t+1)) - 1/(t+2) - 1/(t+3)], since all other terms cancel in pairs = (8/3) (1 + 1/2 + 1/3 - 0) = 44/9. I hope this helps!
Which mathematics topics should I learn to improve my algorithms skills and get started with competitive programming?
Here is the basic math you need to get started with competitive programming :Number Theory - Sieve, primality test, fibonacci etcLinear Algebra - Basic Matrices, multiplication, etcBasic Algebra - Solving linear equations, finding unknown, etcStatistics - Mean, Median, Variance, expected valueBasic Arithmetic concepts like mixtures, progressions, ratios, rational numbers, hcf / lcm etc.Probability - Bayes' theorem, Probability distribution, etcCo-ordinate Geometry - 2D, 3D space, distance between points, line equation, etcDiscrete Mathematics and Combinatorics - Arrangements, sorting, permutationsPrinciples like Pigeon Hole, Inclusion-Exclusion, Induction etcThat is just for starting. Once you are comfortable with competitive programming you can learn following advance topics :Graph Theory - Path, Node, Colouring, Matching, Components, Flow, etcCalculus - Variable Minimisation, MaximisationError Analysis and Estimation. This is important for only few tricky probsGame Theory - Nim Game, Grundy Number, etc.Fast Fourier TranformationModular Arithmetic
If an athlete can run 100 meters in 10 seconds, why would it be mathematically wrong to say that he ran 10 meters in 1 second? That is impossible.
100 meters in 10 seconds is the overall distance and time.an average can be calculated - 10 meters in 1 secondbased on this average it is mathematically correct to say that the athlete ran 10 meters in 1 secondMissing pieces of information:initial accelerationfinal speedNow breaking a sprint down down — there is a period where the athlete:accelerates from [math]0[/math] to some [math]v[/math]maintains the [math]v[/math] till the finish linewe do not care about what happens after the finish linewe could do some of the Kinematic Equations and do some solving to figure it outAlso:Well, I mean running 10 meters in 1 second sounds like a superhero feat and hypothetical, but factually isn’t it correct?! What laws of physics can deny or prove it?Two words: Usain BoltA Kinematics Analysis Of Three Best 100 M Performances EverThis ability, combined with longer stride allows him [Usain Bolt] to create very high running speed - over 12 m/s (12.05 – 12.34 m/s) in some 10 m sections of his three 100 m performances.
Is there a pattern to the prime numbers?
Here's my Prime Number Distribution (founded last night!). Looking for confirmation, or a supercomputer... .First some facts:1. Prime Numbers with 2 or more digits have the sums of their digits equal to Only one of the following - {1,2,4,5,7,8}Example: 11=1+1=2, 23=2+3=5, 103=1+3=4, etc...2. If we group the prime numbers according to their respective sums, the gap in each group between any successive primes - but not restricted to only successive primes in each group - is a multiple of 9.3. All Prime Numbers with two or more digits are located at the intersection of the lines x=1, x=2, x=4, x=5, x=7, x=8 and the lines generated by y=x+9n, where n={1,2,3,...∞}. *** Not all "n" will generate a line such that it's intersection with one of the lines x=1,2,4,5,7,8 will have the y coordinate a Prime Number. Graphs and Some EquationsAs you can see, every line intersects 6 vertical lines. Out of the 6 intersecting points, 3 y coordinates are even numbers and 3 are odd numbers.This reduces the primality test to a maximum 3 numbers to tests in each region. Here's an example:Test for primality 18011, 18013, 18017 . After testing ... 18013 is a prime number.Another Example:329720305014161086772742857 is a prime .(329 septillion 720 sextillion 305 quintillion 14 quadrillion 161 trillion 86 billion 772 million 742 thousand 851)(took a few random tries to find this one - 5 minutes - got luck?)Ask yourself this: How long will it take to find a number in the hundred septillions which turns out to be a prime? To be continued ...
How do you find M in Taylor's Inequality (aka LaGrange Error Bound)?
I know that: |error| =< [ |M|/(n+1)! ] * [ |x-a| ^(n+1) ] |f^(n+1) (x)| =< M but what does M equal? I don't understand what value to put into Taylor's inequality? In the problem I'm doing, I have f(x) = sqrt(x) with a = 4 and n =2 over the interval of [4, 4.2]. I found the Taylor's polynomial to be: 2 + (x-4) / 4 - (x-4)^2 / 64 and that (3/256) =< M but I don't know what to plug into Taylor's inequality. So far I have: |error| = [ |M| / 3! ] / [ |x-4|^3 ]. And where does the interval of [4, 4.2] come into play? With the M? Please help explain this for me. I really want to understand this concept. Thank you so much in advance!