Linear Approximations

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Presentation transcript:

Linear Approximations Implicit Differentiation Linear Approximations

Objectives Students will be able to Calculate the differential of a function Use differentials to approximate values for expressions Use differentials to approximate change in revenue (population, area, volume, and tolerances) Implicit Differentiation Linear Approximations

Vocabulary Linear Approximation Differential Form of the Derivative Implicit Differentiation Linear Approximations

Formulas Volume of a Cube Area of a Circle Volume of a Sphere Volume of a Cone Implicit Differentiation Linear Approximations

Example 1 For the function y below, find dy, given x = -2 and Δx =0.1. Implicit Differentiation Linear Approximations

Example 2 Use the differential to approximate the radical expression Implicit Differentiation Linear Approximations

Example 3 Use the differential to approximate the expression Implicit Differentiation Linear Approximations

Example 4 The demand for grass seed (in thousands of pounds) at a price of p dollars is Use the differential to approximate the change in demand for a change in price from $2 to $2.10. Implicit Differentiation Linear Approximations

Example 5 Beach balls 1 foot in diameter have a thickness of 0.03 inches. How much material would be needed to make 5000 beach balls? Implicit Differentiation Linear Approximations

Example 6 A worker is cutting a square from a piece of sheet metal. The specifications call for an area that is 16 cm2 with an error of no more than 0.01 cm2. How much error could be tolerated in the length of each side to endure that the area is within tolerance? Implicit Differentiation Linear Approximations 10