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10/11/2014 Perkins AP Calculus AB Day 13 Section 3.9

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Linear Approximation Non-calculator application of the tangent line. Used to estimate values of f(x) at ‘difficult’ x-values. (ex: 1.03, 2.99, 7.01) Steps: a.Find the equation of the tangent line to f(x) at an ‘easy’ value nearby. b.Plug the ‘difficult’ x-value in to get a reasonable estimate of what the actual y-value will be.

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1. Find the equation of the tangent line to f(x) at x = 1.

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2. Use the equation of the tangent line to f(x) at x = 1 to estimate f(1.01). This estimate will be accurate as long as the x-value is very close to the point of tangency.

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10/11/2014 Perkins AP Calculus AB Day 13 Section 3.9

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Linear Approximation

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1. Find the equation of the tangent line to f(x) at x = 1.

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2. Use the equation of the tangent line to f(x) at x = 1 to estimate f(1.01).

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Finding Differentials Change in y. Change in x. Slope of tangent line at a given x. 3. Estimate f(0.03) without your calculator. To estimate a y-value using a differential: 1. Find a y-value at a nearby x-value. 2. Add the value of your differential. Differential 4. Estimate f(8.96) without your calculator.

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Finding Differentials 3. Estimate f(0.03) without your calculator. 4. Estimate f(8.96) without your calculator.

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Section 6.1: Euler’s Method. Local Linearity and Differential Equations Slope at (2,0): Tangent line at (2,0): Not a good approximation. Consider smaller.

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