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AP®︎/College Calculus BC
Course: AP®︎/College Calculus BC > Unit 11
Lesson 3: AP Calculus BC 2011- 2011 Calculus BC free response #1a
- 2011 Calculus BC free response #1 (b & c)
- 2011 Calculus BC free response #1d
- 2011 Calculus BC free response #3a
- 2011 Calculus BC free response #3 (b & c)
- 2011 Calculus BC free response #6a
- 2011 Calculus BC free response #6b
- 2011 Calculus BC free response #6c
- 2011 Calculus BC free response #6d
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2011 Calculus BC free response #6c
Calculating the 6th derivative at 0 from the Taylor Series approximation. Created by Sal Khan.
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Is there a difference between the words 'order' and 'degree' when looking at these types of equations?(5 votes)
I'm not sure why you can't just take the 6th derivative of f(x) at 0 by hand especially because all the terms with x just go away. Or am I missing something?(2 votes)
You could, but you would need to contend with multiple applications of the product rule, things can get messy and that is where errors can be made. Doing it that way is often called the "Brute Force" method. Nothing wrong with that.
In this case, the question, and its several sections, are leading you, step by step, to an analytic solution, which is yours for the taking. It is one thing to be able to do the "cookbook" calculations, that is, relying on the application of the rules of differentiation and integration that apply to a given situation, and quite another to start to get a feel for the meaning of what you are doing and the implications of those meanings which will help guide you to analytic solutions.
Don't get me wrong, I am not belittling "Brute Force". You need to be an expert at it and do it many times to start to get a feel for the concepts and implications I am talking about. I am also not belittling you either because you may not yet see the implications, or, if you do, chose not to use that knowledge. Everyone progresses at their own rate, the key is to keep working at it.
Many mathematicians like to joke that they are lazy, but that laziness is an admirable trait because it is actually an optimization of effort, meaning that you can quickly dispense with one problem so as to move on to another.
Great Question!
Keep Studying!(6 votes)
What is the difference between a taylor polynomial and the taylor approximation and the taylor series approximating an insert-the-degree term? Or is there a difference at all?(3 votes)
If instead the question asked what the f(5)(x) derivative value was, would it be equal to 0? As there is no fifth degree term so we can presume it is zero?(2 votes)
Yes, the fifth derivative of f(x) evaluated at 0 would be equal to 0 because there is no fifth degree term.
If you keep taking derivatives of each term in the series, by the time you reach the fifth derivative, the only terms left would be the terms of a higher degree than -121x (the sixth degree term). Because these terms would all equal 0 when x = 0, we know that the fifth derivative is also equal to 0.(2 votes)
Video transcript
Part C. Find the value
of the sixth derivative of f evaluated at 0. So you could imagine
if you just tried to find the sixth derivative of
f, that would take you forever. And then to evaluate it at 0,
because this is x squared here. And you'd have to keep doing
the product rule over and over again, and the chain
rule and all the rest. It would become very,
very, very messy. But we have a big clue here. The fact that they made
us find the first four terms of the Taylor series
of f about x equals 0 tells us that there
might be a simpler way to do this, as
opposed to just taking the sixth derivative of
this and evaluating at 0. The simplest way to do
this is to just go back. In the last problem,
we were able to come up with the first four
non-zero terms of the Taylor series of f. And if you look at your
definition of the Taylor series right here-- and we go into
depth on this in another Khan Academy video where we talk
about why this makes sense-- you see that each degree
term of the Taylor series, its coefficient
is that derivative. And this Taylor Series
is centered around 0, and that's what we care about
in terms of this problem. We see the coefficient is
that derivative divided by that degrees, that
derivative evaluated at 0 divided by that
degrees factorials. So the second degree term,
it's the second derivative of f evaluated at 0
divided by 2 factorial. The fourth degree term
is the fourth derivative of f evaluated at 0
divided by 4 factorial. So the sixth degree term--
let's remind ourselves what we're even
trying to figure out-- so they want us to figure
out the sixth derivative of f evaluated at 0. That's what they want
us to figure out. Well, if you think about
the Taylor series centered at 0, or at 0, or
approximated around 0, the sixth degree term in the
Taylor series approximation of f is going to be f prime
of the sixth derivative of f evaluated at 0 times x to
the sixth over 6 factorial. This is going to
be the sixth degree term in Taylor approximation,
in Taylor series. And we have that term
sitting right over here. This is the sixth degree term. We figured it out
in the last problem. This right here is
the sixth degree term. So you have x to the sixth over
here, x to the sixth over here, you have 6 factorial over
here, 6 factorial over here. So this negative 121 must
be the sixth derivative of f evaluated at 0. So that's our answer. This is equal to negative 121. And we're done.