Math 23

Transcendental Functions

Homework Assignments

HW1 Due Friday, Sept 7. Graph the following functons using the strategy in Chapter 12. Show the chain of transformations and all the intermediate graphs. Don't use your graphing calculuator!!! (You can do it yourself!!!)
1. y=3|x-2|+6
2. y=-(x-1)^2-3
3. y=-(x-5)^3+2
4. y=(1/2)|2x+1|-1
5. y=-(1-2x)^(1/2)
6. x=(3y+2)^2+4
7. y=|(x-5)^2-4|
Note: x^2 means "x squared" and a power of 1/2 means square root.
HW2 Due Monday, Sept 10. Graph the following. These are like the last set, but a litte bit trickier!
1. y=2(x-1)^(1/3)+2
2. y=|(2x+1)^(1/3)|-2
3. (x-2)^2+y^2=9
4. (2x+1)y=0
5. (y-x+1)(y-x^2)=0
6. (2x+1)^2+(y-5)^2=25
7. Bonus problem. This problem is more challenging than our problems will usually be, but it is a really great exercise, so give it a shot.
  A parabola is defined as the set of all points that are equidistant between a given point F and a given line L. The given point F is called the focus and the line L is called the directrix. The point of this exercise is to show that the graph of y=x^2 is indeed a parabola. So we need to find its focus and directrix. Supose the focus is the point (0,a). Then the directrix would HAVE to be the line y=-a. (WHY?) Now write down an equation by equating the distance of the point (x, x^2) to the point (0,a) with the distance from the point (x, x^2) to the line y=-a. Use this equation to find the number a, and hence the focus and the directrix.
HW3 Due Wed, Sept 12. Read Chapter 12. There are 19 problems in the chapter with either solutions or answers. Try to do these problems without looking at the solution and sees how that goes. Even if you first read the problems AND the solutions, then wait a day and then try to do the problems without looking at the solutions. Get a piece of paper and put it over the page in the book and then just slowly pull the paper down to reveal the question without showing the answer. Try it. If this does not work and you want some more exercises, let me know and I can write some up.
HW4 Due Fri, Sept 14. Start reading Chapter 17. Read and understand problems 17.1-17.6. Do these problems and hand them in:
1. Graph (using our transformation technique of Chapter 12) y=2^(x-1)+3.
2. Graph y=3e^(1-2x).
3. Graph y=-2^(x+2).
4. Graph y=e^|x|. (This is different from what we have been doing, but just think about it!)
5. BONUS PROBLEM: Graph y=e^(-x^2). This is again, quite different from what we did in Chapter 12. You can't use the transformation ideas. So think about plotting points, symmetry, and what happens as x gets really big (approaches infinity). Where is the maximum value of the graph and what is it? Try not to use your graphing calculator!
HW5 Due Mon, Sept 17. Keep reading Chapter 17. Do these problems:
1. If you invest $500 at an annual interest rate of 4.5%, compounded annually, how much will you have after 10 years?
2. If you invest $500 at an annual interest rate of 4.5%, compounded monthly, how much will you have after 10 years?
3. If you invest $500 at an annual interest rate of 4.5%, compounded daily, how much will you have after 10 years?
4. If you invest $500 at an annual interest rate of 4.5%, compounded continuously, how much will you have after 10 years?
5. APR vs APY. APR is "Annual Percentage Rate", sometimes called the "nominal rate", while APY is "Annual Percentage Yield". (You might want to take a look at the following website: http://www.investopedia.com/articles/basics/04/102904.asp?partner=answers). In the first four problems, the APR is always the same, namely, 4.5%. But the actual amount you earn is different because the compounding periods were different. The APY is the ACTUAL amount of money your investment earns in 1 year as a percentage of your initial investment. (Note that we used 10 years in the first four problems, so we'd have to go back and redo them to compute the APY for each investment.)
  Compute the APY for each of the following investments to see which is better, that is, has the higher APY.
  APR=5.0% compounded monthly
  APR=4.95% compounded continuously.
6. Assume that when banks compute interest they always round down to the nearest penny. Suppose your savings account earns 4% per year (APR) and is compounded every second. How much money do you have to have in the account so that your earn at least 1 penny every compounding period? What is wrong with the following: If you don't have enough money to earn at least 1 penny every second, then because they round down, you don't earn interest in each compounding period and hence your account never grows at all.
7. Suppose that on the day that he died, Benjamin Franklin invested 1 dollar in your name in a bank account that earns 5% per year compounded continuously. How much is your investment worth today?
HW6 Due Wed, Sept 19. Graph the following cubics by completing the cube!
1. y=x^3-3 x^2+3 x+4
2. y=x^3+3 x^2+2 x+3
3. A good first model of human population growth is to treat the growth similar to the money-in-the-bank problem with continuous compounding. (See exercise 17.13.) Namely, if a population starts out at P and grows with an annual growth rate of r, then the population after t years is Pe^(rt).
  1. If the population of the US is now 300 million and is growing at an annual growth rate of 1.5 per cent, what will the population be in the year 2025? How about 2050? How about 2107?
  2. If the poulation of the earth is now at 6 billion and is growing at an annual rate of 1.7% what will the population be in 2100? How about 2200?
  3. Get on the internet and find some current estimates for a) the population on the earth, and b) the annual growth rate. Using these estimates compute what the population will be in 100, 200, 300, 400, 500, 600, 700, 800, 900, and 1000 years from now.
  4. The radius of the earth is about 4000 miles. What is the surface area of the earth? (You need to know the formula for the surface area of a sphere os radius r.) Take the population estimates you found in the previous problem and estimate the number of people per square mile at each of the times in the future. What do you think of this scenario?
HW7 Due Monday, Sept 24.
1. IF you feel that you need more practice at completing the square and sketching the graphs of quadratics, here are some exercises that you can do. You do NOT need to turn these in.
  1. y=x^2+5x
  2. y=x^2-6x+12
  3. y=-x^2+2x+8
  4. y=3x^2+12x-7
  5. y=-4x^2-10x+4
2. Here are some completing the cube ones that you can practice with too. You do NOT need to turn these in.
  1. y=x^3+12x^2-x-5
  2. y=2x^3+6x^2+x+1
3. Let's compare the exponential model of population growth with the logisitic model. Suppose the population of the earth is now 6.6 billion and growing at an annual growth rate of 1.5%. Use the exponential (unlimited) growth model to predict the population 25, 50, 75, and 100 years from now. Now use the logistic model, with a carrying capacity (P) of 50 billion, to so the same. How do the predictions compare? What if we go out to 200 years from now?
4. Suppose a population is growing exponentially and doubling every 20 years. If it starts out at 6.6 billion, what will it be in 100 years? First write a formula for the population using a base of 2.
5. Radioactive elements appear to decay exponentially. So the amount of time for half of the subtance to disappear depends only on the element, not on how much is present. This is called the half-life. A word that would be more like doubling time would be "halving time." If 10 ounces of Strontium-90 were sitting in the middle of the Pitzer campus, how much would be there 100 years from now? You will need to look up the half-life of Strontium-90 (try the internet.)
6. We have now talked about all the topics that are outlined in Chapter 17. So all of the problems should make sense. I encourage you to read and understand them all! If you want to ask me about your answers to any of them, please drop by my office!
7. Read Chapter 18 and WITHOUT USING A CALCULATOR, compute the following logs. Since I cannot typeset mathematics very easily here, log(2, 8) means "log base 2 of 8"
  1. log(5, 25)
  2. log(3, 27)
  3. log(27, 3)
  4. log(1/2, 1/4)
  5. log(10, .00001)
  6. log(1/4, 64)
HW8 Due Friday, Sept 28. Do exercises 18.13 to 18.22. The answers are there, so you can check to see if you get what they do. READ chapter 19.
HW9 Due Monday, Oct 1. Read Chapter 19. Do the following exercises, which use stuff from all three chapters 17, 18 and 19.
1. How long does it take an invesstment to double if it earns 4.5% APR compounded continously?
2. At what rate (APR) would an investment double in 8 years, if compounded continuously?
3. A good rule of thumb to remember is that a 7% investment will double in 10 years, if continuously compounded? Is this really true? What interest rate is really needed?
4. Suppose you remember, as a rule of thumb, that a 7% investment will double in 10 years? Is it correct to reason that a 14% investment will double in 5 years, or that a 3.5% investment will double in 20 years? If this is true, why? If not, why doesn't it work?
5. Country A presently has 10 million people and an annual growth rate of 1.1%. Country B has a lot more people, 45 million, but is growing slower with an annual rate of only 0.9%. When will the population of B equal that of A?
6. Suppose country C had 17 million people 10 years ago and now had 23.5 million. Assuming that it is growing exponentially, when will it have 35 million?
7. Suppose country D has 8 million people and is growing according to the logistic model with an ultimate cap of 25 million. If in 12 years the population has grown to 9.2 million when will it reach 15 million?
8. Newton's Law of Cooling says that if a hot object at temperature T is placed in a cold environment with ambient temperature A, then the hot object will cool off with T going down closer and closer to A. In fact, the Law asserts that the difference, T-A, will fall exponentially. Suppose a loaf of bread is removed from a 350 degree oven and placed in a room with air temperature 72 degrees. If the bread has cooled to 300 degrees in 5 minutes, when will it reach 100 degrees?
9. Newton's Law of Cooling is used to help fight crime! Suppose a murder victim is found at the bottom of an alpine lake. The water temperature in the lake is a constant 35 degrees. The temperature of the body is measured and found to be 42.5 degrees. Do we have enough information to tell when the vicitm was murdered?
HW10 Due Friday, Oct 5. Do the following:
1. What will be the monthly payment on a car loan of $30,000 at 4.5% ammortized over 5 years? How much do you end up paying in total?
2. Suppose the MOST you can afford on a car loan is $300 per month. If the best loan you can find is 4.0% over 5 years, how much can you afford to borrow?
3. What will be the monthly payment on a mortgage of $450,000 at 6.5% ammortized over 30 years? How much do you end up paying total?
4. Most loans are set up so that you can pay more each month if you want to. How long will it take to pay off the mortage of the previous example if you pay twice the regular payment each month? (Hint: You need the formula that we had for your loan balance P_k at the end of month k as a function of the initial balance, the interest rate and the payment. Use this formula to find out what k is when P_k is zero. Be careful that you answer is a whole number of months. That is, you very last payment may not match each of the ealier payments.) How much do you pay now over the life of the loan?
5. Suppose you have $10,000 cash on hand and want to buy a car for $30,000. The best loan you can find is at 4.5% interest over 5 years. The best investment you can find is a 5 year CD (certificate of deposti) that pays 5.5% interest per year, compounded yearly. Which one of the following scenarios is better (in terms of how much it costs you.)
  - Put the $10,000 down on the car and only borrow $20,000.
  - Put the $10,000 in the CD, and borrow $30,000.
HW 11 Due Friday, Oct 19. Read Chapter 20. Do Exercises 1-24.
HW 12 Due Friday, Oct 26. Read Chapter 21. Do Exercises 1-21.
HW 13 Due Friday, Nov 2. Read Chapter 22. Do all the exercises.
HW 14 Due Monday, Nov 5. Read Chapter 23. Do all the exercises.
HW 15 Due Wednesday, Nov 7. Read Chapter 24. Do 24.5, 247, 24.9, 24.10, 24.12, 24.14, 24.16, 24.17, 24.18, 24.20, 24.21, 24.24, 24.28, 24.32, 24.35, 24.36
HW 16 , Due Friday, Nov 30.