taylor polynomial – Mathemafrica http://www.mathemafrica.org All about maths in Africa Sat, 30 Jul 2016 09:17:05 +0000 en-US hourly 1 https://wordpress.org/?v=4.4.2 UCT MAM1000 lecture notes part 20 – part ii – Monday 17th August http://www.mathemafrica.org/?p=11189 http://www.mathemafrica.org/?p=11189#comments Sun, 16 Aug 2015 08:20:08 +0000 http://www.mathemafrica.org/?p=11189 So, we saw in the last section that we could work out the polynomial expression for (1+x)^5 both using combinatorics as well as using calculus. We had also found previously that for small x we could just take the first couple of terms of the polynomial and it was a good approximation of the function itself, depending on how accurate we needed the answer. For instance, for small x, 1+5x is a reasonable approximation. One thing to note is that the value of these two functions is exactly the same at x=0 and the derivative of both functions is the same at x=0. The second derivatives are not the same, but had we taken the next term in the polynomial, the second derivatives would have matched as well.

OK, let’s take this one step further and repeat everything exactly as before, but now with the power, not as 5 but as 5.2 and see what happens.…

]]> So, we saw in the last section that we could work out the polynomial expression for (1+x)^5 both using combinatorics as well as using calculus. We had also found previously that for small x we could just take the first couple of terms of the polynomial and it was a good approximation of the function itself, depending on how accurate we needed the answer. For instance, for small x, 1+5x is a reasonable approximation. One thing to note is that the value of these two functions is exactly the same at x=0 and the derivative of both functions is the same at x=0. The second derivatives are not the same, but had we taken the next term in the polynomial, the second derivatives would have matched as well.

OK, let’s take this one step further and repeat everything exactly as before, but now with the power, not as 5 but as 5.2 and see what happens.…

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