(a) Find the linear approximation at x=0 to each of f(x)=...

(a) Find the linear approximation at $x=0$ to each of $f(x)=$ $(x+1)^{2}, g(x)=1+\sin (2 x)$ and $h(x)=e^{2 x} .$ Compare your results. (b) Graph each function in part (a) together with its linear approximation derived in part (a). Which function has the closest fit with its linear approximation?

(a) Find the linear approximation at $x=0$ to each of $f(x)=$ $(x+1)^{2}, g(x)=1+\sin (2 x)$ and $h(x)=e^{2 x} .$ Compare
your results.
(b) Graph each function in part (a) together with its linear approximation derived in part (a). Which function has the closest fit with its linear approximation?

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Key Concepts

Taylor Series (First-Order Expansion)

The Taylor series expresses a function as an infinite sum of terms based on its derivatives at a specific point. The linear or first-order approximation is simply the initial part of this series, using only the function's value and first derivative, which is sufficient for local approximations near the point of expansion.

Graphical Interpretation of Approximations

Graphical interpretation involves plotting both the original function and its linear approximation to visually assess the accuracy of the approximation. This comparison helps in understanding how well the linear model captures the behavior of the function around the point of tangency and where discrepancies may arise as you move further from the point.

Linear Approximation

Linear approximation is a method of estimating the value of a function near a given point by using the tangent line at that point. This tangent line is determined by the function's value and its derivative at the chosen point, and it serves as the first-order Taylor expansion of the function.

Derivative

The derivative of a function quantifies its rate of change at any given point. In the context of linear approximation, the derivative provides the slope of the tangent line, which is critical for constructing an accurate local linear model of the function's behavior.

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