Derivatives

Differential Approximation

Differential approximation allows us to estimate numbers that are difficult to calculate by hand. For example, we know $\sin(\pi)$ , but what about $\sin(3)$ ?

Given a function $f$ and a difficult-to-evaluate input $a$ , follow this process to estimate $f(a)$ :

Find a nearby value $a^*$ at which $f$ is easy to calculate
Find the tangent line there, called $L(x)$ , with slope $f'(a^*)$
Estimate $f(a)$ using the height $L(a)$ instead

In point-slope form,

L(x)=f(a^*)+f'(a^*)(x-a^*)

Function

\displaystyle f(x)=\sin x

\displaystyle L(x)=\pi-x

Input to estimate3

\displaystyle f(3)\approx L(3)=0.141593

\displaystyle f(3)=0.14112,\quad |L-f|=0.000473

The approximation is local. Moving farther from the tangency point can increase the error; the tangent line is not the original function.

QUESTION

Which tangent approximation estimates

\displaystyle \sqrt{65}

using

\displaystyle a^*=64

Estimate $\displaystyle \sin(3)$ using differential approximation.

Estimate $\displaystyle \sqrt{65}$ using differential approximation.

◀ Previous SectionLogarithmic Differentiation Next Section ▶The Indefinite Integral

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Derivatives

Differential Approximation

Differential approximation allows us to estimate numbers that are difficult to calculate by hand. For example, we know $\sin(\pi)$ , but what about $\sin(3)$ ?

Given a function $f$ and a difficult-to-evaluate input $a$ , follow this process to estimate $f(a)$ :

Find a nearby value $a^*$ at which $f$ is easy to calculate
Find the tangent line there, called $L(x)$ , with slope $f'(a^*)$
Estimate $f(a)$ using the height $L(a)$ instead

In point-slope form,

L(x)=f(a^*)+f'(a^*)(x-a^*)

Function

\displaystyle f(x)=\sin x

\displaystyle L(x)=\pi-x

Input to estimate3

\displaystyle f(3)\approx L(3)=0.141593

\displaystyle f(3)=0.14112,\quad |L-f|=0.000473

The approximation is local. Moving farther from the tangency point can increase the error; the tangent line is not the original function.

QUESTION

Which tangent approximation estimates

\displaystyle \sqrt{65}

using

\displaystyle a^*=64

Estimate $\displaystyle \sin(3)$ using differential approximation.

Estimate $\displaystyle \sqrt{65}$ using differential approximation.

◀ Previous SectionLogarithmic Differentiation Next Section ▶The Indefinite Integral

Differential Approximation