- 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 [latex]f[/latex] at [latex]x=a[/latex] is given by:
- [latex]L(x) = f(a) + f'(a)(x-a)[/latex]
- Interpretation:
- Uses the tangent line at a point to estimate function values nearby
- Accurate for [latex]x[/latex] close to [latex]a[/latex]
- Based on the idea that smooth functions look linear when zoomed in sufficiently
- Key Formula:
- [latex]f(x) \approx f(a) + f'(a)(x-a)[/latex] for [latex]x[/latex] near [latex]a[/latex]
- Applications:
- Estimating function values
- Root approximation
- Simplifying complex calculations
- Basis for Newton’s method
- Limitations:
- Accuracy decreases as [latex]x[/latex] moves away from [latex]a[/latex]
- Not suitable for functions with sharp turns or discontinuities at [latex]a[/latex]
Find the local linear approximation to [latex]f(x)=\sqrt[3]{x}[/latex] at [latex]x=8[/latex]. Use it to approximate [latex]\sqrt[3]{8.1}[/latex] to five decimal places.
Find the linear approximation for [latex]f(x)= \cos x[/latex] at [latex]x=\dfrac{\pi }{2}[/latex].
Find the linear approximation of [latex]f(x)=(1+x)^4[/latex] at [latex]x=0[/latex] without using the result from the preceding example.
Differentials and Amount of Error
The Main Idea
- For a function [latex]y = f(x)[/latex], the differential [latex]dy[/latex] is defined as: [latex]dy = f'(x) , dx[/latex]
- [latex]dx[/latex] is an independent variable that can be any nonzero real number
- Relationship to Linear Approximation:
- [latex]\Delta y \approx dy[/latex] for small changes in [latex]x[/latex]
- Based on the linear approximation: [latex]f(a + dx) \approx f(a) + f'(a) , dx[/latex]
- Error Estimation:
- Propagated Error: [latex]\Delta y = f(a + dx) - f(a)[/latex]
- Estimated Error: [latex]\Delta y \approx dy \approx f'(a + dx) , dx[/latex]
- Relative and Percentage Error:
- Relative Error: [latex]\frac{\Delta q}{q}[/latex], where [latex]q[/latex] 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 [latex]y=e^{x^2}[/latex], find [latex]dy[/latex].
For [latex]y=x^2+2x[/latex], find [latex]\Delta y[/latex] and [latex]dy[/latex] at [latex]x=3[/latex] if [latex]dx=0.2[/latex].
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.