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
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

Linearization and Newton’s Method

Section 4.5

Warm-Up, Expanded
• Graph each of the following on your graphing calculator:
• 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)