Fall+Week+02

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Monday 8/22
First test of the semester in this class. You will find a copy of the test,along with Ms. Gentry's key, on the @Tests page of this wiki.

Tuesday 8/23
Homework last night was exploratory problem set 2.1 from page 33.

1. The figure below shows the function f(x) =(2x^2 -5x -2) / (x-2) a) Show that f(2) takes the indeterminate form 0 /0 . Explain why there is no value for f(2).

2 for x in both the denominator and the numerator gives zero for both parts. f(2) has no value because division by zero is not defined.

b) The number y = 3 is the limit as x approaches 2 . Make a table of values for each 0.001 unit of x from 1.997 to 2.003. Is it true that f(x) stays close to 3 when x is kept close to 2 but not equal to 2?

We can see that if x is kept within .003 units of 2, f(x) remains within .006 units of 3.
 * x || f(x) ||
 * 1.997 || 2.994 ||
 * 1.998 || 2.996 ||
 * 1.999 || 2.998 ||
 * 2 || error ||
 * 2.001 || 3.002 ||
 * 2.002 || 3.004 ||
 * 2.003 || 3.006 ||

c) How close to 2 would you have to keep x for f(x) to stay within .0001 units of 3? Within .00001 units of 3? How could you keep f(x) arbitrarily close to 3 just by keeping x close enough to 2, but not equal to 2?



If x is kept within .00005 units of 2, f(x) remains within .0001 units of 3. Repeating the procedure on the calculator, zooming in further we can conclude that if x is kept within .000005 units of 2, f(x) remains within .00001 units of 3.

If x is kept, very close to 2, (lets call this distance from 2 the letter delta), f(x) will be kept within twice the distance delta from 3.

d) The missing point at x = 2 is called a __removable discontinuity.__ Why do you suppose this name is used?

The discontinuity can be removed by assigning f(2) = 3. f(x) =(2x^2 -5x -2) / (x-2) can be reduced to f(x) = 2x-1 if x is not equal to 2, by factoring the numerator and reducing common factors. Substituting 2 into 2x - 1 yields 3, the limit of f(x) as x approaches 2.



Thursday 8/25
Definition of limit investigation on the TiNspire. A copy of the handout with accompanying screenshots should appear here soon.

Friday 8/26
Last night we investigated the limit theorems for the sum, difference, product, quotient and composition of two functions. The theorems can be seen in the graphics below.

The essential thing to remember ids that the properties only apply when the limits of functions f and g (the "original" functions) are numbers. You cannot add infinity to infinity, or negative infinity to a constant.



After discussion of these theorems, we worked on a worksheet of problems which required us to use the theorems. These problems can be seen in the slide-show below.

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