Small changes, approximations and errors

What you will be able to do

• use a differential to estimate a small change in a quantity;
• estimate a square root, a cube root or a trigonometric value;
• distinguish absolute, relative and percentage error;
• find the error in an area or a volume from the error in a length.

A small change

Near a point, a curve and its tangent are almost the same. So for a small change $dx$ in $x$, the change in $y$ is close to the change along the tangent:

Here $\Delta y = f(x + dx) - f(x)$ is the true change, and $dy$ is the differential, the change along the tangent. We can rewrite this as an estimate of a value:

The estimate is best when $dx$ is small. The error shrinks very quickly as $dx$ shrinks.

Estimating roots and values

An example with a root. Estimate $\sqrt{66}$. Choose the perfect square nearest to $66$, which is $64$. Let $f(x) = \sqrt{x}$, $x = 64$, $dx = 2$. Then $f'(x) = \dfrac{1}{2\sqrt{x}} = \dfrac{1}{16}$ at $x = 64$, so

The calculator gives $8.12404$, so the estimate is correct to three digits.

An example with an angle. Estimate $\sin 31^\circ$. Work in radians: $31^\circ = 30^\circ + 1^\circ$, and $1^\circ = \dfrac{\pi}{180}$ radians. With $f(x) = \sin x$, $x = \dfrac{\pi}{6}$ and $f'(x) = \cos x$,

An angle must be in radians when you use a derivative of a trigonometric function.

Absolute, relative and percentage error

Suppose a quantity $x$ is measured, and the reading has an error of $dx$.

• The absolute error is $dx$, in the units of $x$.
• The relative error is $\dfrac{dx}{x}$, a pure number.
• The percentage error is $\dfrac{dx}{x} \times 100\%$.

For a quantity $y = f(x)$ calculated from $x$, the absolute error in $y$ is $dy = f'(x)\,dx$, and the relative error is $\dfrac{dy}{y}$.

A very useful rule follows for $y = x^n$. Since $dy = nx^{n-1}\,dx$,

The relative error in $y$ is $n$ times the relative error in $x$. A $1\%$ error in the radius of a sphere gives a $3\%$ error in its volume, because $V \propto r^3$, and a $2\%$ error in its surface area, because $S \propto r^2$.

A worked example

A square metal plate has a side of $20\ \text{cm}$, measured with an error of $0.1\ \text{cm}$. Estimate the error in the area, and find the relative and the percentage error.

1. The link. $A = x^2$, so $dA = 2x\,dx$.
2. The absolute error. $dA = 2 \times 20 \times 0.1 = 4\ \text{cm}^2$.
3. The relative error. $\dfrac{dA}{A} = \dfrac{4}{400} = 0.01$. This is also $2 \times \dfrac{0.1}{20}$.
4. The percentage error. $0.01 \times 100 = 1\%$.

Practise with new numbers

Try it in Python

The program estimates three square roots with a differential and prints the true value and the percentage error each time.

Common mistakes

• Using degrees in a derivative of a trigonometric function. Change the small angle to radians first.
• Choosing a starting number that is not a perfect square. Then $\sqrt{a}$ is not easy to find, and the method has no point.
• Confusing $\Delta y$ and $dy$. $\Delta y$ is the true change. $dy$ is the estimate from the tangent.
• Mixing absolute and relative error. $dx$ has units, and $\dfrac{dx}{x}$ does not.
• Making $dx$ too large. The estimate is only good for small changes.

Remember these

• $dy = f'(x)\,dx$ and $f(x + dx) \approx f(x) + f'(x)\,dx$.
• Absolute error $dx$. Relative error $\dfrac{dx}{x}$. Percentage error $\dfrac{dx}{x} \times 100\%$.
• For $y = x^n$, the relative error in $y$ is $n$ times the relative error in $x$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. For $y = x^2$, find $dy$ when $x = 5$ and $dx = 0.02$.

2. Estimate $\sqrt{37}$ with a differential.

3. A circle has a radius of $10\ \text{cm}$, measured with an error of $0.2\ \text{cm}$. Find the error in its area, and the percentage error in the area.

4. The edge of a cube is measured with a $1\%$ error. What is the percentage error in its volume?

5. Estimate $\sqrt[3]{8.06}$.

References

• NCERT, Mathematics Textbook for Class XII, Part I, Chapter 6 (Application of Derivatives).
• OpenStax, Calculus Volume 1, Section 4.2 (Linear Approximations and Differentials).