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⬅ Back to Differential Calculus Topics

Linear Approximation and Differentials

$$dy = f'(x)\, dx$$
$$\Delta y \approx dy = f'(x)\Delta x$$
$$L(x) = f(a) + f'(a)(x-a)$$

Finding the Differential — CE Board

Find $dy$ if $y = x^3 - 4x + 1$.

The differential is $dy = f'(x)\,dx$.

$f'(x) = 3x^2 - 4$
$\boxed{dy = (3x^2 - 4)\,dx}$

Approximating a Square Root — CE Board

Use differentials to approximate the change in $y = \sqrt{x}$ as $x$ increases from 25 to 26.

$f(x) = \sqrt{x}$, $f'(x) = \dfrac{1}{2\sqrt{x}}$, $a = 25$, $dx = 1$.

$dy = f'(25)\cdot 1 = \dfrac{1}{2\sqrt{25}} = \dfrac{1}{10} = \boxed{0.1}$

So $\sqrt{26} \approx 5 + 0.1 = 5.1$. (Actual: $\sqrt{26} \approx 5.0990$)

Linear Approximation of a Cube Root — CE Board

Approximate $\sqrt[3]{8.1}$ using the linearization $L(x) = f(a) + f'(a)(x - a)$.

Let $f(x) = x^{1/3}$, $a = 8$.

$f(8) = 2,\quad f'(x) = \tfrac{1}{3}x^{-2/3},\quad f'(8) = \tfrac{1}{3\cdot 4} = \tfrac{1}{12}$
$L(8.1) = 2 + \dfrac{1}{12}(8.1 - 8) = 2 + \dfrac{0.1}{12} \approx \boxed{2.0083}$

(Actual: $\sqrt[3]{8.1} \approx 2.00829$)

Error Propagation in Surface Area — CE Board

The radius of a sphere is measured as $r = 6$ cm with a possible error of $dr = 0.05$ cm. Estimate the relative percentage error in the surface area $S = 4\pi r^2$.

$dS = 8\pi r\,dr$

Relative error:

$\dfrac{dS}{S} = \dfrac{8\pi r\,dr}{4\pi r^2} = \dfrac{2\,dr}{r} = \dfrac{2(0.05)}{6} = \dfrac{0.1}{6} \approx 0.0167$
$\text{Percentage error} \approx \boxed{1.67\%}$

Differential of a Product — CE Board

Find $dy$ if $y = x^2 \sin x$.

Apply the product rule:

$\dfrac{dy}{dx} = 2x\sin x + x^2 \cos x$
$\boxed{dy = (2x\sin x + x^2\cos x)\,dx}$
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