Approximations using differentials

A differentiable function near a point looks roughly like its tangent line. This observation gives a quick way to estimate function values without resorting to calculators or tables.

The linear approximation

For differentiable $f$ near $a$, and a small displacement $h$,

$f(a + h) \approx f(a) + h \cdot f'(a).$

The right side is the value at $a + h$ of the tangent line at $a$. The approximation is exact at $h = 0$ and degrades as $|h|$ grows.

Differential notation

In Leibniz notation, $dy = f'(x)\,dx$. The differential $dy$ is the change in $y$ along the tangent line; the actual change is $\Delta y = f(x + \Delta x) - f(x)$. For small $\Delta x$, $\Delta y \approx dy$.

Procedure

1. Identify the function $f$ and the base point $a$.
2. Compute $f(a)$ and $f'(a)$.
3. Choose $h$ so that $a + h$ is the value where we want $f$.
4. Estimate $f(a + h) \approx f(a) + h f'(a)$.

Worked example: estimate $\sqrt{36.5}$

Take $f(x) = \sqrt x$, $a = 36$, $h = 0.5$. $f(36) = 6$, $f'(x) = 1/(2\sqrt x)$, $f'(36) = 1/12$.

$\sqrt{36.5} \approx 6 + 0.5 \cdot 1/12 = 6 + 1/24 = 6.0417$.

Actual value: $6.0415$. Excellent agreement.

Worked example: estimate $\sin 31°$

Convert to radians: $30° = \pi/6$, $1° = \pi/180$. So $31° = \pi/6 + \pi/180$.

Take $f(x) = \sin x$, $a = \pi/6$, $h = \pi/180$. $f(\pi/6) = 1/2$. $f'(\pi/6) = \cos(\pi/6) = \sqrt 3/2$.

$\sin 31° \approx 1/2 + (\pi/180)(\sqrt 3/2) = 0.5 + (0.01745)(0.866) = 0.5 + 0.01511 = 0.5151$.

Actual value: $0.5150$. Within $0.0001$.

Percentage errors

If $x$ is measured with error $\Delta x$, the resulting error in $y = f(x)$ is $\Delta y \approx f'(x)\,\Delta x$. The relative error is $\Delta y / y \approx (x f'(x) / f(x))(\Delta x / x)$, useful in error propagation.

Worked examples

Example 1. Estimate $\sqrt{25.3}$.

$f(x) = \sqrt x$, $a = 25, h = 0.3$. $f(25) = 5, f'(25) = 1/10$. Estimate: $5 + 0.3/10 = 5.03$. Actual: $5.0299$.

Example 2. Estimate $(1.001)^{10}$.

$f(x) = x^{10}$, $a = 1, h = 0.001$. $f(1) = 1, f'(1) = 10$. Estimate: $1 + 10(0.001) = 1.01$. Actual: $\approx 1.01005$.

Example 3. Estimate $\sqrt[3]{0.7}$.

Hmm, base point should be near $0.7$ where cube root is known. $f(x) = x^{1/3}$, $a = 0.729$ (cube of $0.9$), $h = -0.029$. $f(0.729) = 0.9, f'(0.729) = (1/3)(0.729)^{-2/3} = 1/(3 \cdot 0.81) \approx 0.4115$.

$\sqrt[3]{0.7} \approx 0.9 + (-0.029)(0.4115) = 0.9 - 0.0119 = 0.8881$.

Actual: $0.8879$. Close.

Example 4. A square plate of side $5$ cm is heated, increasing each side by $1$ mm. Find the approximate increase in area.

Area $A = x^2$. $dA = 2x \, dx = 2 \cdot 5 \cdot 0.1 = 1$ cm$^2$.

Example 5. Find the approximate change in volume of a cube of side $5$ cm if each side is increased by $2\%$.

$V = x^3$. $dV/V = 3 \, dx/x$, so percentage change in volume is $3 \cdot 2\% = 6\%$. Numerically: $V = 125$, $dV = 0.06 \cdot 125 = 7.5$ cm$^3$.

Example 6. A piece of wire is bent into a square. Find the approximate change in area if the side length increases from $10$ cm to $10.05$ cm.

$A = x^2$, $dA = 2x \, dx = 20 \cdot 0.05 = 1$ cm$^2$.

Try it yourself

1. Estimate $\sqrt{37}$.
2. Estimate $\sqrt[3]{8.06}$.
3. Estimate $(0.999)^{5}$.
4. Estimate $\sin 89°$.
5. Estimate $\cos 60.1°$.
6. Estimate $\tan 46°$.
7. Estimate $\log_e 1.05$ given $\log_e 1 = 0$.
8. The radius of a sphere is measured as $9$ cm with error $0.03$. Find the approximate error in volume.
9. Each side of a cube increases by $1\%$. Find approximate change in surface area.
10. Estimate $\ln 100$ given $\ln 99 \approx 4.595$.
11. Estimate $(15.1)^2$.
12. Estimate $1/\sqrt{0.99}$.
13. Estimate $e^{0.05}$ given $e^0 = 1$.
14. The radius of a circular disc is $10$ cm with error $0.02$. Find percentage error in area.

Pitfalls / Tricks

• The approximation is good only for small $h$. Far from $a$, it can be wildly wrong.
• Choose the base point $a$ where $f(a)$ and $f'(a)$ are easy to compute , typically a clean value like a perfect square.
• For percentage error in a power $y = x^n$: percent error in $y$ is roughly $n$ times percent error in $x$.
• In angle problems, always convert degrees to radians first.

Next, the most important section: maxima and minima.