1. Linear Approximations
Implicit Differentiation
Linear Approximations

2. 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

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

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

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

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

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

8. 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

9. 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

10. 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