Warm up december 19 2012
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Warm-Up: December 19, 2012.

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Warm-Up: December 19, 2012

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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


Homework Questions?


4.1-4.3 Questions?


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


Homework Questions?


Linearization and Newton’s Method

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


Homework Questions?


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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