Find the linear approximation at x=0 to show that the...

Find the linear approximation at $x=0$ to show that the following commonly used approximations are valid for "small" $x$. Compare the approximate and exact values for $x=0.01, x=0.1$ and $x=1$.
$$
e^{x} \approx 1+x
$$

Key Concepts

Linear Approximation

Linear approximation involves using the tangent line at a given point to approximate the behavior of a function near that point. This method simplifies complex functions to a linear form, making them easier to evaluate and analyze when the argument is close to the point of tangency.

Derivative

The derivative of a function at a specific point gives the slope of the tangent line to the function at that point. It is fundamental to linear approximation because the first-order term in the Taylor expansion of a function is determined by its derivative at the approximation point.

Taylor Series

The Taylor series expands a function into an infinite sum of terms calculated from the function's derivatives at a certain point. Truncating this series after the linear term provides a linear approximation, which is especially useful when the variable changes by a small amount.

Exponential Function

The exponential function is a key mathematical function with the property that its derivative is equal to itself. This property makes it an ideal example for demonstrating linear approximation, as its behavior near a point can be accurately captured by simply adding the product of the derivative and the change in the variable.

Error Analysis

Error analysis in the context of approximations involves comparing the approximate value obtained using a linear (or other) approximation with the exact value of the function. This comparison is critical in understanding the limitations and the range of validity for the approximation, particularly as the change from the point of approximation increases.

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