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Warm up december 19 2012
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 20121
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
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
Warm-Up: December 20, 2012

  • Write the equation of the line tangent to




Warm up expanded
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
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
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
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
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
Warm-Up: December 21, 2012

  • Without a calculator, estimate



Newton s method
Newton’s Method

  • Uses linearizations to find the zeros of a function.

  • Process repeats until the answers converge.


Newton s method1
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
Example 3

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


Differentials
Differentials

  • Differentials are like very small deltas

  • Finding a differential is similar to finding a derivative


Example 4
Example 4

  • Find the differential dy.

  • Evaluate dy at x=2, dx=0.1


Example 5
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.


Assignment1
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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