Tangent Line Approximation
Definition
Tangent line approximation, also known as linear approximation or tangent line estimation, is a method that uses the equation of a tangent line at a specific point on a curve to approximate the value of the function near that point. It provides a close estimate when dealing with small intervals.
Analogy
Imagine you're hiking up a mountain trail and want to know how much higher you'll be after taking just one more step. The tangent line approximation is like looking at the slope of your current position on the trail and estimating how much higher you'll be based on that slope.
Related terms
Derivative: The derivative represents the rate of change of a function at any given point and plays a crucial role in tangent line approximation.
Linearization: Linearization is another term for tangent line approximation, where we approximate complex functions with simpler linear functions.
Error Bound: The error bound refers to the maximum difference between the actual value and the estimated value obtained through tangent line approximation.
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Practice Questions (5)
When is the tangent line approximation an underestimate of the actual function value?
How does the tangent line approximation change when the function is concave down?
How can you improve the accuracy of a tangent line approximation?
How does the accuracy of a tangent line approximation change as the interval between points decreases?
What happens to the accuracy of a tangent line approximation as the function becomes more nonlinear?
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