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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 10

Lesson 11: Finding Taylor polynomial approximations of functions

# Worked example: coefficient in 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 coefficient of the x² term in a Maclaurin polynomial, given the formula for the value of any derivative at x=0.

## Want to join the conversation?

• in the fourth term you forget x^3 • While taking the root of 9, Sal said "it's positive 3".
Why does it have to be positive 3 only? If the answer was -(3/16) how will we know? -(3/16) seems like an equally logical answer to me. • The formula which gives you the values of the nth derivative at x=0 really blew my mind, I don't know why but It never crossed my mind that such a thing could exist !
It is really cool, So I have two questions
The first how can I figure out such formulas, is there like some known tricks and methods to figure this out, is it easy should I try it myself or are these tricks and methods complex and I can't figure them out on my own ?
The Second is what kinda of function has this formula ? I want to know the original function that gives you the formula presented in the video, again is there a method and how hard is it , I can't think of anything except graphing the Taylor approximation and even after that the graph didn't seem that familiar, It grew very fast so I guessed factorials and exponential must be involved but that is it.

Thanks for taking the time to read this , I am sorry for making this a long question. • In response to the first question: How can you find a generalized formula for the nth derivative of a function?
You can always represent the nth derivative of a function f(t) as d^n(f(t))/dt^n. Rewriting a general solution for this may not be trivial, but can sometimes be done.
Try taking the first few derivatives and seeing if you can find a pattern. For example, if f(t) = cos(t) you will find that there is a clear pattern in taking the derivatives (this is an easy example if you want to try it). If you can find a pattern, you should be able to write a formula expressing that pattern.
Often there will not be a clear pattern, and in this case it's best to just leave the nth derivative expressed as d^n(f(t))/dt^n.

In response to the second question: Given a formula for the nth derivative of a function, can the original function be found?
The answer is: sometimes yes and sometimes no.
Take the equation given in the video, for example. It's impossible to determine from this equation what g(0) is, since the equation for the nth derivative is only defined for n>=1.
The best you could do for a function like this is to plot a Taylor Series approximation. Note that you can't even plot a true Taylor Series approximation for this example since g(0) must be known, but cannot be determined.

If you continue studying math, you will eventually take a class on differential equations. This is a branch of mathematics that will offer some tools to solve problems like this.
• why do we use the principal root of 9 • At Sal is evaluating the second derivative of g(x). Using the provided equation, this is sqrt(2+7)/2^3. This simplifies to sqrt(9)/2^3.
The "principal root" is another name for the square root. Since he is taking the square root of 9, he referred to the "principle root" of 9.
• 3/8 div 1/2 is 6/8=3/4
if it was mul then it will be 3/16 • I understand all of this, but usually I find all the derivatives first. But that was not done here. Instead it was just the general form g'(x), g''(x),... The derivative forms for sqrt(n+7)/n^3 was never expressed. That step was never executed.

What is my clue in the problem to know I didn't have to make a table of derived forms? Or is it that (n+7)^1/2/n^3 is already the derived function?

Because my first instinct was to start listing all the derivations of (n+7)^1/2/n^3
(1 vote) • Almost everytime you write a Taylor series, you would be required to find the derivatives. But, you don't have to here. Why? Because you can't.

Observe the question tackled in the video. You don't have a function to take a derivative for. What's given is the general formula for the derivative itself. So, if you wanted the first derivative, plug in n=1. If you wanted the 59th derivative, plug in n=59. And so on. You can't take the derivatives of the given function as n there is a constant. So, taking the first derivative itself would end up giving 0.

So, your clue is essentially this: See if you have a function to even take the derivative of
(1 vote)
• Is it very common to get question like this, say on exams or tests? I cannot really tell what is being asked in the example.
(1 vote) • let's say I want to find the complete maclaurin expansion of g(x) for the given question. Is the given info enough, or do we need a little more to solve it?
I have a feeling we need g(0) provided to us, but I'm not sure whether there's some clever way of finding that from the given question.
(1 vote) • Why do you not have to take the second derivative to find the solution to this problem?
(1 vote) • The function in the question `g^n(0)=sqrt(n + 7)/n^3` actually gives you the derivatives for the function g at x = 0; whatever you plug in for n will result in that order of derivative for the function g at x = 0. If you plugged in 1 for n, you would get g(0), if you plugged in 3, you would get g^3(0), etc. Normally, you would have to manually take the derivative twice, but in this case, you would just have to plug in 2, like in the video.
``g^n(0)=sqrt(n + 7)/n^3`` 