Approximating Values (Edexcel A-Level Mathematics): Revision Notes

• Example:
• $0.004567$ to three significant figures is $0.00457$
• $123,456$ to three significant figures is $123,000$

• Formula: If you have a function $f(x)$ and you want to approximate it near a point $a$, you use:

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

where $f'(a)$ is the derivative of $f(x)$ at $x = a$

• Example: Approximating $\sqrt{4.1}$ :
• $f(x) = \sqrt{x}$, and we know $f(4) = 2$ and $f'(x) = \frac{1}{2\sqrt{x}}$
• So, $f'(4) = \frac{1}{4}$
• $\sqrt{4.1} \approx 2 + \frac{1}{4} \times 0.1 = 2 + 0.025 = :success[2.025]$

Example: Approximate $(1.02)^5$ :

Example: Approximate $e^x$ near $x = 0$ :

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

For $x = 0.1$ :

$e^{0.1} \approx 1 + 0.1 + \frac{(0.1)^2}{2} = 1 + 0.1 + 0.005 = :success[1.105]$

Example

Question

1. Expand $(1 - 2x)^{\frac{1}{2}}, |x| < \frac{1}{2}$, in ascending powers of $x$ up to and including the term in $x^3$.
2. By substituting a suitable value of $x$ in your expansion, find an estimate for $\sqrt{0.98}$.
3. Show that $\sqrt{0.98} = \frac{7}{10} \sqrt{2}$ and hence find the value of $\sqrt{2}$ correct to 8 significant figures.

Solution:

a. Expand $(1 - 2x)^{\frac{1}{2}}, |x| < \frac{1}{2}$, in ascending powers of $x$ up to and including the term in $x^3$.

b. By substituting a suitable value of $x$ in your expansion, find an estimate for $\sqrt{0.98}$.

Notice $(1-2x)^{\frac{1}{2}} \equiv \sqrt{1-2x}$, so think about what x should be to get 0.98 under the square root.

Checking this answer using a calculator is very close, therefore this gives a good approximation.

c. Show that $\sqrt{0.98} = \frac{7}{10} \sqrt{2}$ and hence find the value of $\sqrt{2}$ correct to $8$ significant figures.

From $(b): \sqrt{0.98} \approx 0.9899495$

Summary:

• Rounding: Adjusting to a nearby value.
• Truncation: Cutting off digits without rounding.
• Significant Figures: Keeping a specified number of meaningful digits.
• Linear Approximation: Using the tangent line at a point to approximate the function.
• Binomial Approximation: Simplifying expressions like $(1 + x)^n$ for small $x$
• Taylor Series: Expanding a function into a series to approximate its value.