If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Worked example: Maclaurin polynomial

AP.CALC:
LIM‑8 (EU)
,
LIM‑8.A (LO)
,
LIM‑8.A.1 (EK)
,
LIM‑8.A.2 (EK)
,
LIM‑8.B (LO)
,
LIM‑8.B.1 (EK)
Finding the second degree Maclaurin polynomial of 1/√(x+1).

Want to join the conversation?

  • blobby green style avatar for user Chou Dunzhi
    I still confuse about the function of X in Taylor formula.
    This formula uses for estimate at point A, so what is the X use for?
    (6 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user ashwinranade99
      a is the point where you base the approximation, but you can vary x in order to get an approximation of the function itself using the polynomial.
      For example, f(a) = P(a) (because you know the value of a), but f(x) ~ P(x) (because P(x) gets you an APPROXIMATION of f(x))
      (8 votes)
  • piceratops seed style avatar for user Pedro Lopes
    What kinds of functions are exactly the same as their taylor series and what kinds of functions are not? In other words, what are the conditions for a function to equal its taylor infinite sum?
    (7 votes)
    Default Khan Academy avatar avatar for user
  • aqualine tree style avatar for user Teddy
    When writing out the Taylor/Maclaurin series, what is the purpose of dividing each term by a factorial? How does that make the polynomial a better approximation as opposed to not dividing by the factorial? He probably answered this in the intro videos but I may have missed it or simply didn't understand.
    (4 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user RowanH
      It comes from the fact that you differentiate several times to get eg the second or third derivative. If you know the value you want eg. the third derivative to have, you need to work out what original term would give you that value when differentiated 3 times.
      Say you know at the point you are centering you the third derivative is a, then the original coefficient for the term in the polynomial to give that would be a/(3*2*1). Try for a Maclaurin series: a/(3*2*1) * x^3.
      differentiate once: a/(2 * 1) * x^2
      differentiate second time: ax
      differentiate third time: a
      (4 votes)
  • piceratops tree style avatar for user piotr1212
    Why would one want to approximate the function with a polynomial at a specific x if you could just fill in x in the original function?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • male robot johnny style avatar for user Sidhusan Devamanoharan
    we take the principle root for f(0), because the function is defined for positive root , isnt it ?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • aqualine seedling style avatar for user Orest Karpiak
    How can you find limit to 0 and to positive infinity using Taylor's polynomial formula?
    like: lim(x->0) of [ (a^x-6)/(x^3)] ?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • winston baby style avatar for user Broken Fixer
      Limit to 0 can be found by replacing functions with their Taylor polynomials to high enough degree to evaluate directly plugging in x=0. So (cos(x)-1) / x^2 would be approximated by (-x^2/2 +x^4/4! -...) / x^2 = - 1/2 + x^2-stuff = -1/2 as x->0.
      (1 vote)
  • blobby green style avatar for user satish chander jain
    what is [1-(-1)^n] as n goes to infinity
    (1 vote)
    Default Khan Academy avatar avatar for user
  • starky ultimate style avatar for user Nordin Shafiq
    Does it make sense to have a Taylor polynomial for a function at it's vertical asymptote line?
    For Example: A Maclaurin polynomial for f(x)=1/x
    (1 vote)
    Default Khan Academy avatar avatar for user
  • leafers seed style avatar for user Vanessa Wojtalewicz
    Whats the final function?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user Alexandra P.
    Hello, I am just learning calculus and Sals's videos are really helpful.
    I can't understand why p(x)= f(0) + f(0)x + f`(0)x^2/2...

    Is it because we are integrating the function at f(0)? Is that why those coefficients and powers show up?
    Thanks
    (1 vote)
    Default Khan Academy avatar avatar for user

Video transcript

- [Instructor] We're told that f of x is equal to one over the square root of x plus one, and what we want to figure out is what is the second degree Maclaurin polynomial of f? And like always, pause this video and see if you could have a go at it. so let's remind ourselves what a Maclaurin polynomial is, a Maclaurin polynomial is just a Taylor polynomial centered at zero, so the form of this second degree Maclaurin polynomial, and we just have to find this Maclaurin expansion until our second degree term, it's going to look like this. So p of x, I'm using p for polynomial, it's going to be our f of zero plus, and we could do that as f of zero times x to the zero power, well, that's just f of zero. F of zero plus f prime of zero x, plus f prime prime of zero, divided by, we could think of it as two factorial, but it's really just two, we could think of this as dividing by one factorial, which is just one, this dividing by zero factorial, well that's just one again, so we have f prime prime of zero, the second derivative evaluated zero divided by two, x squared. Now if we wanted a higher degree we could keep on going, but remember, they're just asking us for the second degree. So this is the form that we're gonna need, we're gonna have these three terms. So let's see if we can evaluate the function and its derivatives at zero. So f of zero, f of zero is equal to one over the square root of zero plus one, well that's one over the square root of one, the principle root of one, which is positive one, so that's just going to be equal to one. So that right over there is equal to one. Now let's evaluate f prime of x, and then I'll evaluate f prime of zero. F prime of x is equal to, well, one over the square root of x plus one, this is the same thing as x plus one, let me write it this way, this is the same thing as, actually let me write it down this way. Another way of writing f of x is this is the same thing as x plus one to the negative one half. And so if I'm thinking the first derivative of f, well, I could use the chain rule here, the derivative of x plus one with respect to x, well, that's just going to be one, and then I'll take the derivative of this whole thing with respect to x plus one and I'll just use the power rule there, this is gonna be negative one half, times x plus one, to the, and I decrement the exponent, negative three halves. And so the first derivative evaluated of zero is just negative one half, times, if this is zero, zero plus one is just one, one to the negative three halves, one to the negative three half power, well, that's just going to be one. So this whole thing, f prime of zero is just negative one half. So that is this right over here is negative one half. And now let's figure out the second derivative. All right, let me do this in this green color. So the second derivative with respect to x, well I did the same thing again, the derivative of x plus one with respect to x, that's just one, so I just have to take the derivative of the whole thing with respect to x plus one. So I take my exponent, bring it out front, negative three halves times one half, times negative one half, is going to be positive three fourths times x plus one, and then I decrement the exponent here by one, or by two halves, so that's gonna be negative five halves, and so the second derivative evaluated at zero, well if this is equal to zero you're gonna have one to the negative five halves which is just one, times three fourths, is gonna be three fourths. So this part right over here is three fourths. And so you're gonna have three fourths divided by two, three fourths divided by two is three eighths. So our Taylor, I should say, our Maclaurin polynomial, our second degree Maclaurin polynomial, p of x is going to be equal to, and I'll do it in the same colors, going to be equal to one, plus, maybe I'll just write it as minus one half, minus one half x, plus three eighths x squared, plus three eighths x squared, and we are done, there you have it. We have our second degree Maclaurin polynomial of f, which could be used to provide an approximation for our function, especially if our x is near zero.