(10) What is the domain of log 123 (x)+ x

Transcription

1 EXAM 1 MASTER STUDY GUIDE MATH 131 I don't expect you to complete this. This is a very large list. I wanted to give you as much information about things that could be asked of you as possible. On Tuesday, I will choose a few (3 or 4) specic problems to go over and then I will ask y'all for your questions. This document is best utilized by marking which items would be EASY for you, which would be MEDIUM, for you, and which would be DIFFICULT for you. This will give you an idea of which types of things you need to work on, practice, and ask me questions about. You should also review your Webassign problems. Your test will be some multiple choice/true false and some work out. More than half of it will be work out and partial credit will be given. You do not need to bring a scantron. There will be a seating chart. Bring you ID, your calculator, and a pencil. I believe in you. You can do the thing.-taylor Just keep swimming-dory 1. Lesson 1.1 (1) Simplify (x + y) 2. Know this. (2) Give me four dierent ways you can represent a function. (3) Represent the function f(x) = x 2 with domain { 2, 1, 0, 1, 2} via each of these ways. (4) Know the dierence between the intervals (0, 1],(0, 1),[0, 1], and [0, 1). (5) Explain what the vertical line test is. (6) Give an example of a graph that is not the graph of a function. (7) What is the domain of f(x) = 1 x? (8) What is the domain of log 2 (x)? (9) What is the domain of 1 x + log 2(x)? (10) What is the domain of log 123 (x)+ x 13 x? (11) Know what it means for a function to be even, odd, increasing, or decreasing. (12) Is f(x) = x 2 increasing on the interval (, 0) or (, 0]? 1

2 2 EXAM 1 MASTER STUDY GUIDE MATH 131 (13) Understand the dierence between an independent variable and a dependent variable. (14) Is 4 in the range of f(x) = x 4 + x 2? (15) *Write an example of an even function. Write an example of an odd function. 2. Lesson 1.2 (1) I drive 50 miles in a straight line at a constant speed of 62 miles per hour. How long was I driving? Is my distance from my starting point to my destination a linear function with respect to time? Model this linear function (nd a function which describes my distance with respect to time). (2) * I can write a test consisting of 20 questions in 5 hours. I can write a test consisting of 40 questions in 7 hours. Assuming that the number of questions I can write is linear with respect to time, nd a function which describes how long it takes me to write a test with x many questions on it. What does the slope represent? How long would it take me to write a test with 100 quesiton on it? (3) Know whether or not a function is polynomial, a power function, a root function, algebraic, trigonometric, exponential, or logarithmic. Give an example of each. (4) Know whether or not a bunch of data given (as points plotted on an xy plane) is appropriately modeled by a line. 3. Lesson 1.3 (1) *Write two of your favorite functions down. Call them f(x) and g(x). Write down what (f g)(x) and (g f)(x) are. (2) *Given a function like x can you write it as the composition of two functions? (3) Given a function, (for example, f(x) = x 2 ), write the formula for the function whose graph is the graph of f(x) shifted to the right 4, then ipped over the y axis, then shifted to the left 3. Try another combination of translations and reections. (4) What do I have to do to f(x) to ip its graph over the x axis and then shift it π units up? (5) If f(x) = x 2 and g(x) = 6 x, what is the domain of (f g)(x)? What is the domain of (g f)(x)? (6) *Write 4 of your favorite functions down. Call them f(x), g(x) and h(x). Compute (f f g)(x) and (f g h g)(x). You do not have to simplify.

3 EXAM 1 MASTER STUDY GUIDE MATH Lesson 1.5 (1) What is an exponential function? Give a specic example. What is its domain? What is its range? (2) Remember: if f(x) = a x then a has to be positive. (3) Know what the base of an exponential function is. (4) *What is x 5 x 1 9? (5) What is (x 5 ) 2? (6) What is x 5 x 4 ((x 2 ) 2 ) 2? (7) Plot f(x) = ( 1 x. 2) (8) *Simplify ( x 3 y 5 y 1 xy ) 2. (9) If a population doubles every 5 years and at year zero the population is 100, what is the function describing the population with respect to time (in years)? How long until the population reaches 1111 people? (10) The concentration of a substance in water halves every hour. At t = 0 hours, the concentration is 10mg. How long until the concentration is 1.2mg? (11) What does the graph of the function f(x) = e look like where e is Euler's number? 5. Lesson 1.6 (1) What does it mean for a function to be one-to-one? Give an example of a function that is one-to-one. Give an example of a function that is not one-to-one. (2) Write down your favorite even function. Is it one-to-one? Why or why not? (3) Can a function which is not one-to-one have an inverse? (4) Draw a function represented as an arrow diagram that is one-to-one. Draw one as an arrow diagram that is not one-to-one. (5) *What is the inverse of the function f(x) = 2 x? (6) What is the inverse of f(x) = x ? (7) Does f(x) = x 2 have an inverse? (8) What is the inverse of the function f(x) = e x? (sometimes this is called the natural logarithm). (9) Know the rules which govern logs and exponential functions (like the fact that log a (xy) = log a (x) + log a (y)). (10) If log a (x) = 14 and log a (y) = 10 what is log a ( axy y 3 )?

4 4 EXAM 1 MASTER STUDY GUIDE MATH 131 (11) What is the inverse of the function f(x) = log 2 (x) + 1? (12) *What is the inverse of the function f(x) = 1 x 2+x? What is its domain? What is its range? (13) Solve the equation log 2 (x + 5) = 10. (14) Solve the equation e x = 2. (15) Solve the equation 3 2x = 4 (16) *Solve the equation e 14x 3 15 = 0. (17) Solve the equation (3 x ) = 0. (18) Is there a solution to 12 x = 10? (19) Is there a solution to e x = 1? (20) Suppose f(x) has domain (, ) and range (, ) f 1 (x) is its inverse. What is the domain and range of f 1 (x)? What is f(f 1 (x))? What is f 1 (f(x))? 6. Lesson 2.1 (1) Describe, in words, what a tangent line is. (2) Describe, in words, what a secant line is. (3) Let f(x) = x Pick a point on this graph (like P = (0, 5)) and another (say Q = (2, 9)). Compute the slope of the secant line connecting P and Q. (4) *Remember that a point P is on the graph of the function f(x) if and only if P = (x, f(x)) for some x. THIS IS IMPORTANT. (5) Let P = (0, 5) and let Q be the point on the graph of f(x) = x 2 x with x coordinate equal to a. What are the coordinates of Q? (it will be in terms of a). (6) The distance a ball has fallen in meters after t seconds is given by 4.9t 2. Describe how to nd the average velocity between times t = 0 and t = 1. (7) Remember this phrase The average velocity is equal to the slope of the secant line of the distance function. (8) Remember this phrase The average velocity is equal to the change in position over the change in time (9) Remember this phrase The instantaneous velocity at t = a is equal to the slope of the TANGENT line of the distance function at a (10) Understand all of those phrases.

5 EXAM 1 MASTER STUDY GUIDE MATH Lesson 2.2 (1) Write down the denition of a limit. (2) *Let Graph this function. f(x) = 1 x (0, 1] x 2 x (1, 3) 3x + 2 x [3, 4) 0 x = 4 x + 10 x (4, 5) Compute the one sided limits of the above function at the points 0, 1, 3, 4 and 5. Compute the value of f(x) for x equal to 0, 1, 3, 4 and 5 individually. (3) Write down some of your favorite functions and compute limits. (4) Draw some graphs of functions and compute limits. (5) Remember that a limit exists at a for f(x) if and only if the one sided limits of f(x) at a exist and are the same. (6) When does a limit not exist? Look at the examples in the notes. 8. Lesson 2.3 (1) Suppose that lim x a f(x) = 3 and lim x a g(x) = 5 and lim x a h(x) = 0. (a) What is lim x a (f + g + h)(x)? (b) How about lim x a (fgh + g)(x)? ( ) (c) How about lim f x a h (x)? (d) Come up with some other combination like this try to solve it. (2) What is the limit lim h 0 h 2 +h h?