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Warm-Up: December 19, 2012

Warm-Up: December 19, 2012.

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Warm-Up: December 19, 2012

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  1. 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.

  2. Warm-Up: December 19, 2012 • Rectangular pool • Area = 1800 sqft • 5 ft deck on sides • 10 ft deck on ends • Minimize property area

  3. Homework Questions?

  4. 4.1-4.3 Questions?

  5. 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

  6. Warm-Up: December 20, 2012 • Write the equation of the line tangent to

  7. Homework Questions?

  8. Linearization and Newton’s Method Section 4.5

  9. 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

  10. 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)

  11. Example 1 – page 229 #4 • Find the linearization L(x) of f(x) at x=a • How accurate is the approximation

  12. 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.

  13. 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)

  14. Warm-Up: December 21, 2012 • Without a calculator, estimate

  15. Homework Questions?

  16. Newton’s Method • Uses linearizations to find the zeros of a function. • Process repeats until the answers converge.

  17. 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

  18. Example 3 • Use Newton’s method to estimate all real solutions of the equation. Make your answers accurate to 6 decimal places.

  19. Differentials • Differentials are like very small deltas • Finding a differential is similar to finding a derivative

  20. Example 4 • Find the differential dy. • Evaluate dy at x=2, dx=0.1

  21. 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.

  22. 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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