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junior college 2 | H2 Maths
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Good afternoon! Can someone please help me with the second part of this qns? I don’t really understand what they mean by “x is small enough for powers of x above the third to be neglected”.. Thanks so much:D
Date Posted: 4 years ago
Views: 330
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Good afternoon Sonia! Here are my workings for this question. We simply replace the “x” in the original expression by “x + x2”.
Remember binomial theorem? The one which goes like
Expand (1 + x)^9 up to the term in x3 and hence use your result to write down the first four terms in the expansion of (1 + x + x^2)^9.
It’s a very similar idea.
Remember binomial theorem? The one which goes like
Expand (1 + x)^9 up to the term in x3 and hence use your result to write down the first four terms in the expansion of (1 + x + x^2)^9.
It’s a very similar idea.
Date Posted:
4 years ago
Sonia, this expansion is valid for small values of x. This is because the remaining terms in the expansion will have higher powers of x.
When x is smaller than 1, the higher the power of x, the smaller the value of the expression.
So for example, if x = 0.01, then 0.01^10 is definitely much smaller than 0.01^2. The contribution by these higher powers of 0.01 will be way too small for the actual value of the number itself, which is why the higher powers in the x can be considered to be neglected when x is small.
Sonia, if you let x = 0.01 in the expansion to evaluate 1/sqrt 1.0101, you will notice that your estimated value from the expansion is going to be very close to the actual value of the expression, because the remaining terms are way too small in quantity to contribute to the sum. It’s like saying, $100000 is a significant sum of money to a millionaire, but give him 5 cents more and he does not become MUCH richer.
When x is smaller than 1, the higher the power of x, the smaller the value of the expression.
So for example, if x = 0.01, then 0.01^10 is definitely much smaller than 0.01^2. The contribution by these higher powers of 0.01 will be way too small for the actual value of the number itself, which is why the higher powers in the x can be considered to be neglected when x is small.
Sonia, if you let x = 0.01 in the expansion to evaluate 1/sqrt 1.0101, you will notice that your estimated value from the expansion is going to be very close to the actual value of the expression, because the remaining terms are way too small in quantity to contribute to the sum. It’s like saying, $100000 is a significant sum of money to a millionaire, but give him 5 cents more and he does not become MUCH richer.
Thanks so much for your fast response! :)
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