Do you know how to find the distance between two points?

Transcription

1 Some notation to understand: is the line through points A and B is the ray starting at point A and extending (infinitely) through B is the line segment connecting points A and B is the length of the line segment between A and B Do you know how to find the distance between two points? We had a quiz on 10/4 after which we briefly covered sections 3A & 3B. Go through the HW well on 10/8 Powers of Review and understand exponent notation 3A: #1,3,6,7 (Exponential notation) 3B: #11 last 4, 12 (Exponent laws) Thinking of this as 1 times 3, five times or (1)(3)(3)(3)(3)(3) may help you to understand negative exponents (later). Can your understanding of exponents help you evaluate 2 4? +1 means multiply 1 by a, n times +1 Zero Exponent Property: a 0 = 1 for all a 0 means divide 1 by a, n times One of the biggest issues with powers is being clear what the base is. Evaluate The exponent operates on the symbol that immediately precedes it! 10/4 3A Exponents 1

2 1. Review and understand the origins of various exponent laws and properties Using your understanding of exponents, write down an equivalent power for each product. Using your understanding of exponents, write down an equivalent power for each quotient. Using your understanding of exponents, write down an equivalent power for each power. Using your understanding of exponents, write down an equivalent power for each power. Using your understanding of exponents, write down an equivalent power for each power. 10/4 3B Laws of Exponents 2

3 There are a lot of useful properties. Do not memorize them! Understand them! Properties of Exponents Let a and b be real numbers and m and n be integers. Then: For now, we will restrict ourselves to integer exponents. More on that next time... This takes practice... 3A: #1,3,6,7 (Exponential notation) 3B: #11 last 4, 12 (Exponent laws) 10/4 3B (cont) Summary & Practice 3

4 We will continue at a fast pace. Hang in there and keep up with your homework. The quiz on this chapter will be Monday 10/22. Might seem far away, but it'll come fast. It will include a "speed quiz" of powers of 2 (up to 2 10 ) and perfect cubes (up to 10 3 ). Begin to memorize them now! PSAT Review 10/8 PSAT Review 4

5 Return Function Quizzes Know your powers of 2 (up to 2 10 ) and perfect cubes (up to 10 3 ). Begin to memorize them now! 3A: #1,3,6,7 (Exponential notation) Present #6,7 3B: #[1,2,5,81] last 3, 12 (Exponent laws) Practice previous page. Questions? 3C: #1 5 last 4 (Rational exponents) 3D.1: #1 2 last col (Algebraic expansion) 1. Understand and apply rational exponents Note change from green HW plan handout We have discussed integer exponents. Today we'll look at rational exponents (ie. fractional exponents). Consider, for example, the exponent of one half. Recall that we can think of exponents as the number of times that you multiply 1 by a base, say means multiply one by 3, five times. So what would it mean, for example, to multiply one by 3, a half a time? Can we support this using properties we know? The product of powers property can help. It implies that But the number that multiplies by itself to get a already has a name: So... and in general which is the nth root of a. The properties that we have developed imply some other interesting things: What happens if we cube the square root of 16? What happens if we take the square root of 16 cubed? Let's look at that using exponents: means the (square root of 16) cubed...or the square root of (16 cubed) In general, Rational Exponents Some practice: Write as single powers of 2 or 3 Try a couple on your calculator: 10/9 3C Rational Exponents 5

6 Expand expressions involving integer and rational exponents Recall the common properties of algebraic expansion. Guess what you can do this with exponents but you need to understand conceptually what factors are and how exponents work. Some examples: Careful!!! Also know how to deal with different situations: Practice is critical! Work thoroughly, make no move without absolutely knowing why. Pay attention to the pattern (not the answers). Because next time we are going in the other direction (aka factorisation). You try some: 3C: #1 5 last 4 (Rational exponents) 3D.1: #1 2 last col (Algebraic expansion) HW is not a lot of practice. Do more problems until you are getting them correct the first time - 1 minute per problem maximum! 10/9 3D.1 Expansion 6

7 Warm up: Simplify the expressions. Simplify the expressions. Rational Exponent Quizzes 7

8 Know your powers of 2 (up to 2 10 ) and perfect cubes (up to 10 3 ). Solidify your memory now! 3C: #1 5 last 4 (Rational exponents) Verbal quiz on all 3D.1: #1 2 last col (Algebraic expansion) Present 1i,2fil Factor expressions involving integer and rational exponents 3D.2: #1 5 last row, 6 (Factorization) 3E: #1 5 last row (Exponential equations) QB: #1,2 (QB: Exp Equations) Last time we expanded. This time, we will do the reverse of that... factor. Let's review: A huge skill, as important as dividing (it actually is dividing) Factoring means writing an expression as a product of factors. In essence we are undoing multiplying. Look for common factors in each term. 6x + 18 = 6(x + 3)...but not just numbers! Common factors can be variables! You may remember some like this: The same ideas hold with variables in the exponent: You may be able to factor an expression even if it has no common factors! There are several methods to try: 1) Reverse FOIL Guess and check The British method (for quadratics with a not equal to one) 2) Recognizing a special pattern 3) Factor by grouping Special Factoring Patterns A Key Idea: a 2x = (a x ) 2 which we can treat the same way we would treat blah 2 In SL, these ideas get extended in many directions, including with exponents: (3 x + 2)(3 x 2) (5 + 2 x )(5 2 x ) (3 x + 5) 2 Question: Why? would it be useful to know how to factor these things? Simplifying Writing an expression in an equivalent form We also need to know how to simplify expressions involving exponents: 4 x 1 = (2 x + 1)(2 x 1) HW is not a lot of practice. Do more problems until you are getting them correct the first time 1 minute per problem maximum! 10/11 3D.2 Factoring 8

9 1. Solve exponential equations by equating exponents Exponential expressions often occur in equations that we need/want to solve. We've seen this a little bit before. In general, we may need to use logs. But we'll save that for the next chapter. Meanwhile, notice that there are some forms of exponential equations that we can solve without using logs. Suppose, for example, that we wanted to solve the equation 8 = 2 n Because this can be written as 2 3 = 2 n we can mathematically support what we know to be true. Equating Exponents If you can rewrite an exponential equation to have powers of the same base on either side, you can equate the exponents to solve the equation. Try a few Another idea: Notice the quadratic form. Factor and then find each zero. A few more. Notice how you can manipulate things. You're looking for: > One term on each side > The same base on each side (rewrite the bases) > Expressions in the exponents that you can set equal to each other > Also look for quadratic forms > There may not be solutions This is the minimum homework. Do enough to become proficient. 3D.2: #1 5 last row, 6 (Factorization) 3E: #1 5 last row (Exponential equations) QB: #1,2 (QB: Exp Equations) The quiz on this chapter will be Monday 10/22 It will include a "speed quiz" of powers of 2 (up to 2 10 ) and perfect cubes (up to 10 3 ). Memorize them now! 10/11 3E Exponential Equations 9

10 The quiz on this chapter will be next Monday 10/22 It will include a "speed quiz" of powers of 2 (up to 2 10 ) and perfect cubes (up to 10 3 ). Know them forward and in reverse! 3D.2: #1 5 last row, 6 (Factorization) Present 1f,2hi,3ef,4gh,5i,6ab 3E: #1 5 last row (Exponential equations) Take questions QB: #1,2 (QB: Exp Equations) Present QB Quick Quiz 1. Write as a single power of an appropriate base. [2 marks each] A2 3 (3 2a) A2 2. Expand and simplify fully. [4 marks each] 3 2x + 7(3 x ) + 10 A1A1A1 9 x + 7(3 x ) + 10 A1 7 2x x A1A1A1 49 x 2 + 1/49 x A1 1. Graph and understand graphs of exponential functions. 3F: #1bd,2 6all (Exponential functions) 3G.1: #2,4 (Exponential growth) Graphs help us to visualize relationships (functions). What does an exponential look like? Start with a table of values: x /2 2 1/4 Notice that the function can never reach zero. It has an asymptote at y=0. What is the range of the function? {y y > 0} What is the affect of changing the base? GGB Demo How about the parameter a? The general exponential has four parameters that control its shape: Understanding this, we can graph exponential functions. To graph: > Plot the y intercept > Plot two other points (x = 1 and x = are usually easy) > Know the general shape and connect To find the function from a graph is not so easy because there are so many possibilities. 10/16 3F Graphs of Exponentials 10

11 Understand and use principles of exponential growth and decay Imagine that 100 rabbits are dropped onto an island and they immediately start reproducing at the rate of 50% per year. Can you write a function that gives the population of rabbits at some number of years n after the initial drop off? Can you graph the function? Increasing a number by p% is equivalent to multiplying the number by 1 + p/100 Year Calculation Population at year's end (1.5) (1.5)(1.5) (1.5)(1.5)(1.5) n 100(1.5)(1.5)...(1.5) 100(1.5) n The population after the n th year is given by the function P(n) = 100(1.5) n Let's look at the graph of this function: Wow This is the equivalent of 3 child families (assuming no deaths, of course!) But you get the picture. More realistic: World population growth rate is approximately 1.17% per year: Here's the same data in another view. What does it communicate? Consider this context: So ask yourself why some understanding of math is important... 3F: #1bd,2 6all (Exponential functions) 3G.1: #2,4 (Exponential growth) 10/16 3G.1 Exponential Growth 11

12 Quiz next time (Monday). It will include a "speed quiz" of powers of 2 (up to 2 10 ) and perfect cubes (up to 10 3 ). Know them forward and in reverse! Most of you would be wise to do some more work on sections 3C 3E. Your quick quizzes showed basic understanding but careless mistakes under pressure. Do a lot of problems as fast as you can. 3F: #1bd,2 6all (Exponential functions) Present 4 (what did you learn?), 6abcd 3G.1: #2,4 (Exponential growth) 1. Understand and use principles of exponential growth and decay 3G.2: #1,3,4,5 (Exponential decay) 3H: #1,3,6,7b,103 (The number e) Review 3C: #11 as needed Exponential Growth The amount after t years of something growing at a rate of r % per year is given by: A(t) = A 0 (1 + r) t where A 0 is the initial amount present. r is called the growth rate and is written as a decimal. The value 1 + r is called the growth factor. This idea can be extended for any time period as long as t and r are given in the same units of time (years, months, seconds, etc.) Exponential functions also describe decay. A common example is radioactive decay where, for example, a radioactive substance deteriorates (decays) at a rate of 5% per year. Now we have a "growth multiplier" (1 + r) that is less than 1. It's really (1 r). The amount of the substance after n years is given by W n = W 0 (0.95) n Suppose we started with 50 grams. We can ask questions such as: How much is present after 10 years? When will the amount left be equal to than 10 grams? (we'll leave that for later!) Radioactive materials are often described by their half life. This is the amount of time it takes to reach half the initial weight. Given that some material has a half life of 1000 years, what is the annual rate of decay? Thus the annual decay rate is = =.0693% The exponent is not always an integer number like n. Let's look at some examples: V = P(1 0.07) t 3000 = P(0.93) /(0.93) 3 = $3, /18 3G.2 Exponential Decay 12

13 1. Understand the origins of the number e. 2. Use e in continuous growth and decay problems. Let's go back to compound interest again. You will recall that the formula for the amount of money in the bank after t years if you invest P dollars at an annual interest rate of r, compounded annually is given by: A = P(1 + r) t compounded annually If you compound, say, quarterly, you will be paid 4 times as often but the interest rate you receive each quarter is only a fourth of the annual interest rate. Thus, the formula becomes: If you compound monthly, there are 12 periods. So the formula becomes: A question arises: what if you compounded "continuously" meaning that we let the number of periods per year get larger and larger on to infinity! Let's look and see if there's a pattern. To do this, we need to generalize our equation first. Let's use n to represent the number of compoundings. Then we have: To explore this, let's make a substitution. Can you show algebraically that This is nice because we can explore the factor in square brackets. Notice that as n gets larger and larger, so does a (it's just n divided by a constant). So let's see what happens as a gets large. a This number is the natural number called "e" after Leonhard Euler who discovered it. It shows up in nature in many strange and interesting ways. It also is the sum of It is irrational! So...when we have continuous compounding (or continuous growth) a gets very large and our equation: becomes A(t) = Pe rt Continuous Exponential Growth The amount after t years of something growing continuously at a rate of r % per year is given by: A(t) = Pe rt where P is the initial amount present. r is the growth rate (r > 0) or the decay rate (r < 0) and is the decimal % change in one unit of time (t). Question: Suppose you have 10 grams of Somestuffium that grows continuously at a rate of 50% per year. How much will you have after one year? Your friend has 10 grams of Otherjunkium that decays continuously at a rate of 50% per year. How much will your friend have after one year? You: 10e (0.5)(1) = 10e 0.5 = ~16.5 grams Your friend: 10e ( 0.5)(1) = 10e 0.5 = ~6.07 grams Spread this HW out! 3G.2: #1,3,4,5 (Exponential decay) 3H: #1,3,6,7b,103 (The number e) Review 3C: #11 as needed There will be a quiz on Exponents, powers of 2, and perfcect cubes next time 10/18 3H The number e 13