Linear Approximations and Differentials: Fresh Take

Explain and use linearization to approximate a function’s value near a specific point
Calculate and interpret differentials to estimate small changes in function values
Measure the accuracy of approximations made with differentials by calculating relative and percentage errors

Linear Approximation of a Function at a Point

The Main Idea

Linear approximation (or tangent line approximation) of f at x=a is given by:
- $L(x) = f(a) + f'(a)(x-a)$
Interpretation:
- Uses the tangent line at a point to estimate function values nearby
- Accurate for $x$ close to $a$
- Based on the idea that smooth functions look linear when zoomed in sufficiently
Key Formula:
- $f(x) \approx f(a) + f'(a)(x-a)$ for $x$ near $a$
Applications:
- Estimating function values
- Root approximation
- Simplifying complex calculations
- Basis for Newton’s method
Limitations:
- Accuracy decreases as $x$ moves away from $a$
- Not suitable for functions with sharp turns or discontinuities at $a$

Find the local linear approximation to $f(x)=\sqrt[3]{x}$ at $x=8$ . Use it to approximate $\sqrt[3]{8.1}$ to five decimal places.

$L(x)=f(a)+f^{\prime}(a)(x-a)$

$L(x)=2+\frac{1}{12}(x-8)$ ; 2.00833

Find the linear approximation for $f(x)= \cos x$ at $x=\dfrac{\pi }{2}$ .

$L(x)=f(a)+f^{\prime}(a)(x-a)$

$L(x)=−x+\frac{\pi}{2}$

Find the linear approximation of $f(x)=(1+x)^4$ at $x=0$ without using the result from the preceding example.

$f^{\prime}(x)=4(1+x)^3$

$L(x)=1+4x$

The Main Idea

For a function y=f(x), the differential dy is defined as: dy=f′(x),dx
- $dx$ is an independent variable that can be any nonzero real number
Relationship to Linear Approximation:
- $\Delta y \approx dy$ for small changes in $x$
- Based on the linear approximation: $f(a + dx) \approx f(a) + f'(a) , dx$
Error Estimation:
- Propagated Error: $\Delta y = f(a + dx) - f(a)$
- Estimated Error: $\Delta y \approx dy \approx f'(a + dx) , dx$
Relative and Percentage Error:
- Relative Error: $\frac{\Delta q}{q}$ , where $q$ is the actual value
- Percentage Error: Relative error expressed as a percentage
Applications:
- Approximating function value changes
- Estimating measurement errors in calculations
- Analyzing accuracy in scientific measurements

For $y=e^{x^2}$ , find $dy$ .

$dy=f^{\prime}(x)dx$

$dy=2xe^{x^2} \, dx$

For $y=x^2+2x$ , find $\Delta y$ and $dy$ at $x=3$ if $dx=0.2$ .

$dy=f^{\prime}(3) \, dx$ , $\Delta y=f(3.2)-f(3)$

$dy=1.6$ , $\Delta y=1.64$

Estimate the error in the computed volume of a cube if the side length is measured to be 6 cm with an accuracy of 0.2 cm.

$dV=3x^2 \, dx$

The volume measurement is accurate to within $21.6 \, \text{cm}^3$ .