In Exercises 13-20, using Linear Approximation, estimate Δf for a change in x from x=a to x=b. U

step 1 For this problem, we are asked to use linear approximation to estimate delta f for a change in x

In Exercises 13-20, using Linear Approximation, estimate Δf...

In Exercises 13-20, using Linear Approximation, estimate $\Delta f$ for $a$ change in $x$ from $x=a$ to $x=b$. Use the estimate to approximate $f(b)$, and find the error using a calculator.
$$
f(x)=\ln x, a=1, b=0.97
$$

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Key Concepts

Linear Approximation

Linear approximation uses the tangent line at a point on a function to estimate the value of that function near the point. This method relies on the idea that for small changes in the input, the function behaves nearly linearly, so the tangent line provides a good approximation to the function.

Derivative

The derivative of a function represents its instantaneous rate of change at a specific point. In linear approximation, the derivative at the point of interest is used as the slope of the tangent line, which determines how the function changes for small increments in the input.

Differentials

Differentials refer to the small changes in the function value and the input variable. They are used to estimate the change in the function using the product of the derivative (or slope of the tangent line) and the change in the input, providing a practical means to approximate function values near a known point.

Error Estimation

Error estimation involves comparing the value obtained from a linear approximation to the exact function value. This step is crucial for assessing the accuracy of the approximation and understanding the limitations of linear models when dealing with non-linear functions.

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