Learn how to use linear approximation to approximate the value in calculus. Practice differentials & relative error & linear approximation examples in calculus.

We define differential \(dx\) as an independent variable such that differential dy is defined as

\(dy=f'(x)\;dx\) whenever \(y=f(x)\).

- \(dy\) is dependent variable which depends on \(dx\) and \(x\).
- The idea of differentials is to find out the change in dependent variable \(dy\) when there is a change in the independent variable \(dx\) for a certain function \(f(x).\)

\(\Delta y=\)Actual change in y

\(dy=\) amount by which tangent line rises or falls.

A \(\dfrac {1-x}{2\sqrt x{(1+x)}^2}\;dx\)

B \(\dfrac {1+x}{(1-x)^2}\;dx\)

C \(\dfrac {1-x}{2{(1+x)^2}}\;dx\)

D \(2\,tan\,x\;dx\)

Consider a curve \(y=f(x)\)

A point \(P(x,f(x))\) and \(Q(x+\Delta x,\;f(x+\Delta x))\)are take on it.

We see that \(\Delta y = f(x+\Delta x)-f(x)\)

whereas \(dy=f'(x)dx\)

A \(dy=-0.5 \\ \Delta y=-0.48\)

B \(dy=-0.025 \\ \Delta y=-0.023\)

C \(dy=0.1 \\ \Delta y=0.09\)

D \(dy=-1.2\\ \Delta y=-1.3\)

Suppose volume of sphere

\(V=\dfrac {4}{3}\pi r^3\), then if there is an error of \(\Delta r\) or \(dr\) in measurement of radius then the maximum error in the measurement of volume is

\(d V=4\,\pi\,r^2\;dr\)

- Relative Error = \(\dfrac {d V}{V}=\dfrac {4\,\pi\,r^2\;dr}{4/3\,\pi\,r^3}=\dfrac {3}{r}\;dr\)
- Whenever two quantities are related an error in measurement of one quantity will lead to an error in measurement of other. Differentiate both sides of equation between two quantities.

A 240 cm3

B 510 cm3

C 105 cm3

D 70 cm3

Sometimes for a complicated function, we do not know or it is difficult to find the value at a particular value of x, in these cases we use the tangent line at\( (a,\, f(a))\) as an approximation to the curve \(y = f(x) \)where x is near 'a'.'.

Consider a curve \(y = f(x) \) and points \(A\, (a,\, f(a))\) and \(B\, (b,\, f(b))\) on it.

\(\therefore\) Slope of chord \(AB=\dfrac {f(b)-f(a)}{b-a}\)

If B is close to A or \(b-a\rightarrow0\), we can say that slope of tangent at

\(A\cong\dfrac {f(b)-f(a)}{b-a}\)

- Now let \(b=a+h\), where 'h' is small.

\(\therefore \;\dfrac {dy}{dx}\Bigg|_{x=a}\cong\dfrac {f(a+h)-f(a)}{h}\)

\(\therefore f(a+h)\cong f(a)+h\dfrac {dy}{dh}\Bigg|_{x=a}\)

- We say that

\(L(x)=f(a) + (x-a) f'(a)\) (a + h = x)

is the linearization of 'f' at a.

- We can approximate the values of \('f'\) at values of x close to \('a'\) by this approximation formula.

Steps for finding the linearization of 'f' at x = a

1. Find f '(x)

2. Use \(L(x)=f(a) + (x-a) f'(a)\)

(a will be given)

- \(L(x)\) gives the approximate value of function when x is close to 'a'

A \(\dfrac {7x}{4}-\dfrac {1}{5}\)

B \(\dfrac {7x}{6}+12\)

C \(\dfrac {8x}{3}-\dfrac {16}{3}\)

D \(\dfrac {8x}{5}+\dfrac {1}{4}\)

The function \(y=f(x)\) can be evaluated for dy, using

**\(dy=f'(x)\;dx\)**

Relative Error = \(\dfrac {\text {Maximum Error}}{\text {Total quantity}\nearrow}(\text {whose error is measured})\)

Suppose the radius of sphere is measured and found to be 'a' with maximum positive error of say 't' then

\(V=\dfrac {4}{3}\,\pi\,r^3\)

\(\Rightarrow dV =4\pi\,\underbrace{r^2}_{a^2}\;\underbrace{dr}_{t}\) ( \(dV \rightarrow\) approximate to error in volume = \(\Delta V\))

\(=4\pi\,a^2\,t\)

- If errors are expressed in percentage, such errors are called percentage errors.
- For example

Relative Error = \(\dfrac {\text {Measured Value - True Value}}{\text {True Value}}\)

- Percentage Error = Relative error × 100
- Suppose the radius of the sphere is \(r\) then its volume is \(V=\dfrac {4}{3}\,\pi\,r^3\)
- If the error in the measured value of \(r\) is denoted by \(dr\).
- The relative error in the radius is given by \(\dfrac {dr}{r}\)
- Percentage error will be given by \(\dfrac {dr}{r}×100\)