1. 3.9 Differentials
Linear Approximations and Applications:
Determining the “effect of a small change”.
Examples:
How does a small change in angle affect the distance of a basketball shot?
How are revenues at the box office affected by a small change in ticket prices?
The cube root of 27 is 3. How much larger is the cube root of 27.2?

2. So, using a linear line,
Linear Approximation = slope of the tangent line

3. Example 1: Use the Linear Approximation to estimate
Graphical view:
Example 1: Use the Linear Approximation to estimate
How accurate is your estimate?

4. Differential Notation: The Linear Approximation to y = f(x) is often written using the “differentials” dx and dy. In this notation, dx is used instead of
∆x to represent the change in x, and dy is the corresponding vertical change in the tangent line:

5. Example 2: How much larger is than ?
Example 3: A thin metal cable has length L = 12 cm when the temperature is T = 21oC. Estimate the change in length when T rises to 24oC, assuming that

6. Example 4: The Domina Pizza Company claims that
its pizzas are circular with diameter 50 cm.
What is the area of the pizza?
Estimate the quantity of pizza lost or gained if the diameter is off by at most 1.2 cm.

7. Linearization: Approximating f(x) by its Linearization
Example 5: Compute the linearization of

8. Example 6: Estimate and compute the percentage error of