Bailey World of Math

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Class 2019-2020 EasyClass Webpage Calendar & Pacing Guide
     
 AP Calculus AB  AP Calculus AB 
Calculus Honors   Calculus Honors  
Intro to Programming Intro to Programming 

Leaves and Wombats Scenario

Since there aren't answers to the p.68 "Questions to Guide your Review", I will post them here:
1. The real numbers lines are represented by the number line. The main categories characterizing properties of real numbers are algebraic, order and completeness properties. The primary subsets are the natural numbers, the integers, the rational numbers, and the real numbers.
2. The rational numbers have repeating decimals. The irrational numbers don't have repeating decimals.
5. If c is the center of an interval and r is the radius of the interval, the interval can be represented by |x-c|< r or |x-c|<=r. If c is the center of the union of two intervals and r is the radius, then the interval is represented by |x - c | >=r
16. A function is a relation in which each x-coordinate has only one y-coordinate. The domain of a function is the set of all x-coordinates. The range of a function is the set of all y-coordinates. An arrow diagram has an arrow from each input to each output of a function.
20. A function is increasing if a<b means that f(a) < f(b). A decreasing function is a function in which a < b means that f(a) > f(b). Graphically, if a function is increasing it is sloping upward (from left to right) and if the function is decreasing then it is sloping downward (from left to right).
23. If f(x) + g(x) exists, then f(x) and g(x) must exist. That means that each number in the domain of f+g must be in the domain of f and the domain of g. So, the domain of f+g, f-g and fg are the intersections of the domain of f and the domain of g. The domain of f/g is the intersection of the domain of f and the domain of g minus the values at which g(x) = 0
28. A radian is the arc length in the unit circle subtended by the central angle, A.
To convert from radians to degrees you multiply by 180 and divide by pi.
To convert from degrees to radians you multiply by pi and divide by 180.

Hint for Prob #1: ******* This will definitely be on the test ************************

If you look at our function translation notes:
-f(x) is reflected over the x-axis, f(-x) is reflected over the y-axis.
f(x)+D will shift f(x) vertically by D and
f(x-H) will shift f(x) horizontally by H (note the minus sign
Af(x) will "stretch"(expand if A>1/compress if A<1) f(x) vertically by A
f(Bx) will "stretch"(expand if A>1/compress if A<1) f(x) horizontally by 1/B

Hint Prob #2: You are trying to come up with functions that will give you that
f(g(x)) = g(f(x)). You may have to try a few, but there are some very simple ones. Remember when you were in algebra 2 and you had to create functions which were directly or inversely proportional?

Hint Prob #6: Come up with 2 or 3 examples of odd functions and see if you can find a pattern for the value at x = 0. Use the algebraic definition of odd to prove your guess. Yes, you have to prove it for the extra point. That means a word or 2.

Hint Prob #19: So, to make sure you read the problem, you are using the following definition: a < b iff b - a > 0
So to show that a + c < b + c, you would have to show that (b+c) - (a+c) > 0 provided that b - a > 0 (because a < b)
To show that ac < bc, you will need to show that bc - ac < 0, again provided that b-a>0 and c > 0.

Hint Prob #24: *************This is my favorite problem in this series. ******
They give you a hint. They tell you that no matter what function f(x) is, (f(x) + f(-x))/2 is an even function. If I were you, I would try it. Graph f1(x) = x^3 + 2x^2 - 1in the calculator. It is neither even nor odd. However, if you enter f2(x) = ( f1(x) + f1(-x) )/2, you will see that it is even (or symmetric over the y-axis). equivalently, f3(x) = ( f(x) - f(-x) )/ 2 will be an odd function. If you add f1(x) and f2(x) together, what do you get? I will give you 1 point for part (a) and 1 point for part (b). You need to SHOW that this is always true. That means you can't use an example to prove it. However, if you do use an example to "prove" the statement, just go back and replace the example function with f(x), and you will likely have a proof.
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