Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. It is particularly useful for simplifying complex functions and provides an accurate estimate when the function is continuous and differentiable at the point of interest.

Linear Approximation

Linear approximation

method used to estimate the value of a function

A tangent line to a curve at a given point is a straight line that just touches the curve at that point, having the same direction as the curve's slope there. It is used in calculus to approximate the curve near that point and is fundamental in understanding instantaneous rates of change and derivatives.

tangent line

simplifying complex functions

accurate estimate

A continuous function is one where small changes in the input lead to small changes in the output, ensuring there are no sudden jumps or breaks in its graph. Continuity is a fundamental property in calculus and analysis, crucial for understanding limits, derivatives, and integrals.

Continuous Function

A differentiable function is a function whose derivative exists at each point in its domain, indicating that it has a well-defined tangent line at every point and is smooth without any sharp corners or discontinuities. Differentiability implies continuity, but the converse is not necessarily true, as a function can be continuous without being differentiable at certain points.

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