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Warm-Up: December 19, 2012.

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### Linearization and Newton’s Method

Warm-Up: December 19, 2012

- A rectangular swimming pool is to be built with an area of 1800 square feet. The owner wants 5-foot wide decks along either side and 10-foot wide decks at the two ends. Find the dimensions of the smallest piece of property on which the pool can be built satisfying these conditions.

Warm-Up: December 19, 2012

- Rectangular pool
- Area = 1800 sqft
- 5 ft deck on sides
- 10 ft deck on ends
- Minimize property area

4.1-4.3 Quiz

- Clear everything off of your desk except pencil and eraser.
- NO CALCULATOR!
- 20 minute time limit
- You must remain silent until all quizzes have been turned in.
- If you finish early, reread Section 4.5

Warm-Up: December 20, 2012

- Write the equation of the line tangent to

Section 4.5

Warm-Up, Expanded

- Graph each of the following on your graphing calculator:
- Start with a standard window
- Zoom in at the origin repeatedly and observe what occurs

Linearization

- If f is differentiable at x=a, then f is locally linear.
- Zooming in very close, f looks like a straight line.
- The linearization of f at a is:
- The approximation f(x)≈L(x) is the standard linear approximation of f at a.
- (Related to Taylor Series – Calculus BC topic)

Example 1 – page 229 #4

- Find the linearization L(x) of f(x) at x=a
- How accurate is the approximation

Example 2 – page 229 #12

- Choose a linearization with center not at x=a but at a nearby value at which the function and its derivative are easy to evaluate. State the linearization and the center.

Assignment

- Read Section 4.5 (pages 220-228)
- Page 229 Exercises #1-13 odd
- Page 229 Exercises #15-35 odd
- Read Section 4.6 (pages 232-236)

Warm-Up: December 21, 2012

- Without a calculator, estimate

Newton’s Method

- Uses linearizations to find the zeros of a function.
- Process repeats until the answers converge.

Newton’s Method

- Step 1: Guess an approximate root/zero/x-intercept, x1
- Step 2: Use the first approximation to get a second approximation
- Use the second approximation to get a third, the third to get a fourth, and so on

Example 3

- Use Newton’s method to estimate all real solutions of the equation. Make your answers accurate to 6 decimal places.

Differentials

- Differentials are like very small deltas
- Finding a differential is similar to finding a derivative

Example 4

- Find the differential dy.
- Evaluate dy at x=2, dx=0.1

Example 5

- Write a differential formula that estimates the change in surface area of a sphere when the radius changes from a to a+dr.

Assignment

- Read Section 4.5 (pages 220-228)
- Page 229 Exercises #1-13 odd
- Page 229 Exercises #15-35 odd
- Read Section 4.6 (pages 232-236)

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