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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
Area = 1800 sqft
5 ft deck on sides
10 ft deck on ends
Minimize property area
Clear everything off of your desk except pencil and eraser.
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
Graph each of the following on your graphing calculator:
Start with a standard window
Zoom in at the origin repeatedly and observe what occurs
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.
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
Uses linearizations to find the zeros of a function.
Process repeats until the answers converge.
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
Use Newton’s method to estimate all real solutions of the equation. Make your answers accurate to 6 decimal places.
Differentials are like very small deltas
Finding a differential is similar to finding a derivative
Find the differential dy.
Evaluate dy at x=2, dx=0.1
Write a differential formula that estimates the change in surface area of a sphere when the radius changes from a to a+dr.