General Binomial Expansion (AQA A-Level Mathematics): Flashcards

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Approximating Values
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What is truncation?

Cutting off digits without rounding up

Round 5.67895.6789 to 2 decimal places

5.685.68

0.0045670.004567 to 3 significant figures

0.004570.00457

Linear approximation formula for f(x)f(x) near aa

f(x)f(a)+f(a)(xa)f(x) \approx f(a) + f'(a)(x - a)

What does linear approximation use?

Tangent line at a point on curve

Binomial approx. of (1+x)n(1 + x)^n for small xx

(1+x)n1+nx(1 + x)^n \approx 1 + nx

When does (1+x)n1+nx(1 + x)^n \approx 1 + nx work?

When xx is small

What is Taylor series approximation?

Expansion into infinite sum of terms from derivatives

Taylor approx. of exe^x near x=0x = 0 (first 3 terms)

ex1+x+x22!e^x \approx 1 + x + \frac{x^2}{2!}

Truncate 5.67895.6789 to 2 decimal places

5.675.67

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