Math 121. Practice Problems from Chapter 4 Fall 2016

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1 Math 11. Practice Problems from Chapter Fall 01 Section 1. Inverse Functions 1. Graph an inverse function using the graph of the original function. For practice see Eercises 1,.. Use information about the original function to find things such as values, domain, range of inverse function. For practice see Eercises 3,. 3. Given a function, find its inverse function and find domains and ranges of the function and its inverse. For practice see Eercises 5,, 7.. Check if two functions are inverses of each other using the composition propert. For practice see Eercises, 9. Section. Eponential Functions 1. Solve eponential equations with the same base. For practice see Eercise 1.. Graph an eponential function, and then graph related functions (such as inverse, shifted, reflected, etc). Eercises, Given the graph of an eponential function. Find its base, and graph related functions (such as inverse, shifted or reflected, etc) find its inverse function and find domains and ranges of the function and its inverse. For practice see Eercises, 5,.. Determine information based upon eponential models. For practice see Eercises 7,. 5. Know the natural base e. See Eercise 9.. Use properties of eponents to deal with large numbers. See Eercise 10. Section 3. Logarithmic Functions 1. Evaluate logarithms b converting to eponential form. For practice see Eercise 1.. Find domains of logarithmic functions. Eercises, Convert eponential equations to logarithmic form and vice versa. For practice see Eercises, 5.. Graph logarithmic functions using eponential functions. Then graph translations, etc of logarithmic functions. For practice see Eercises, Graph logarithmic functions using eponential functions, b converting to eponential form. For practice see Eercises, 9.. Identit a base from the graph of a logarithm function, and then graph related functions. See Eercises 10 and Convert a common logarithm to scientific notation. See Eercises 1 and 13.. Describe in words how various logarithmic and eponential graphs relate to each other. See Eercise 1

2 Section. Properties of Logarithms 1. Use properties of logarithms to epand logarithms. For practice see Eercises 1,.. Use properties write logarithmic epressions as a single logarithms. For practice see Eercises 3,. 3. Use properties of logarithms to evaluate logarithmic epressions. For practice see Eercise 5.. Applications of logarithms. For practice see Eercises (Richter Scale), 7 (Decibel Levels), (ph), 9 (Radiation Penetration). 5. Use logarithms to manipulate large numbers (see Eercises 10, 11, 1). Section 5. Eponential and Logarithmic Equations 1. Solve eponential equations. For practice see Eercises 1,, 3,.. Solve logarithmic equations. For practice see Eercises 5,, Solve a population growth application problem. For practice see Eercise. Section. Applications of Eponentials and Logarithms. 1. Interest income problems. For practice see Eercise 1.. Eponential growth problems. For practice see Eercise, 3, 3. Radioactive deca problems. For practice see Eercise 5,.. Newton s law of cooling. For practice see Eercise 7,. Page

3 1 Inverse Functions 1. The graph of a function f is given below. On same graph sketch the inverse function of f; notice that f goes through the points (0, ) and (1, 1). f. A function f that is one-to-one on its domain is graphed below. For our reference, the points plotted on the graph of f are (, 5), (3, ), (0, 3), ( 5, ) (a) Sketch the graph of f 1 on the same graph. (b) Find (i) f 1 ( ) and (ii) f 1 ( 5). (c) Find (i) the domain of f 1 and (ii) the range of f Suppose f is a function and g is its inverse function, suppose also the domain of f is [1, 17] and the range of f is [ 15, 9] with f() = and f(11) = 1. (a) Is f one-to-one? (b) Is g one-to-one? (c) Find the domain and range of g. (d) If possible, find: g(), g(), g(11), g(1). Page 3

4 . Suppose f() = (a) Evaluate: f( 1), f(0), f(1), f(). (b) Given that f has an inverse function, use our answers from (a) to find f 1 (103), f 1 (1), f 1 () and f 1 ( ). 5. Find the inverse function of f() = Once ou have found f 1 (), verif that f(f 1 ()) = for all.. (a) Find the inverse function of f() = 3 + 1, and find its domain. 7 (b) Find the range of f. (c) Solve the equation =, if possible. Note. This is just asking ou to solve 7 f() = for. (d) Solve the equation = 3, if possible. 7. Let f() = 5 +. (a) Find f 1 (). (b) Find the domain of f. (c) Find the domain of f 1. (d) Find the range of f. (e) Find the range of f 1.. Use the composition propert of inverses to determine if f and g defined below are inverses of each other. ( + f() = 1 9 and g() = 1 ) Use the composition propert of inverses to determine if f and g defined below are inverses of each other. f() = 10 and g() = 1 10 Page

5 Eponential Functions 1. Use properties of eponents to solve = (calculators should not be used).. Consider the function f() = (a) Complete the following table of values for f (b) Sketch a graph of f, and on the same coordinate aes sketch = and the graph of f 1 () (c) Using our answer from (b), sketch g() = Consider the function f() = ( 1 3 (a) Complete the following table of values for f. ) (b) Sketch a graph of f, and on the same coordinate aes sketch = and the graph of f 1 () (c) Using our answer from (b), sketch g() = ( 1 3) 3.. The graph of an eponential function = b is given on the graph below. (a) Use the graph to estimate b. Page 5

6 (b) On the same graph, graph = b The graph of an eponential function = b is given on the graph below. (a) Use the graph to estimate b. (b) Using the graph from (a), graph = b. (c) Using our graph from (b), graph = b +1.. The graph of an eponential function = b is given on the graph below (a) Use the graph to estimate b. (b) On the same graph, graph = b. 7. The number of bass in a lake is given b P (t) = e 0.07t Page

7 where t is the number of months that have passed since the lake was stocked with bass. (a) How man bass were in the lake immediatel after it was stocked? (b) How man bass were in the lake 1 ear after it was stocked? Round our answer to the nearest whole number. (c) What will happen to the bass population as t increases without bound?. The population of a small cit is currentl and is growing at 5 percent per ear. Thus the population is given b P (t) = 70000(1.05) t where t is time measured in ears from the present. (a) What will the population of the cit be in one ear? (b) According to this model, what will the population of the cit be in 1 ears from now? Epress answer to the nearest whole number. (c) Suppose Charles has an investment account that is growing a a rate of 5 percent per ear, and he currentl has dollars in the account. How much mone will be in the account 1 ears from now? Epress answer to the nearest dollar. 9. The number e and the natural eponential and logarithm functions. ( (a) The number e is defined b e = lim n. Use our calculator to complete the n n) following table to get an idea of the approimate value of e: ( Value of n Value of ) n n 10 (1.1) 10 = 100 (1.01) 100 = ( ) = Value of e obtained b using e function on calculator (with = 1): (b) Complete the following table using the e function on our calculator (use decimal places). Then sketch a rough graph of f() = e e Page 7

8 (c) On the graph given in (b) sketch the following functions: (i) g() = e 3 (ii) h() = e 3 (iii) k() = log e Hints. (i) This is a horizontal translation of the graph of f() = e. (ii) This is a vertical translation of the graph in (i) (iii) log e is usuall denoted b ln and called the natural logarithm. k() = ln() is the inverse function of f() = e. The function 10. Eponential functions are ideal for dealing with large numbers (or etremel small numbers), because once we know the base, onl the eponent changes. For eample and are numbers larger than man calculators will accept, et properties of eponents make them eas to multipl (or divide). Indeed, = = Properties of eponents also make it eas to write the above answer in scientific notation = = Use properties of eponents to answer the following questions. (a) Write in scientific notation. Use significant figures in our final answer. (b) Find the product as a power of 10. (c) Convert our answer in (b) to scientific notation. Use significant digits in our answer. Page

9 3 Logarithmic Functions 1. (a) Evaluate log 1. (b) Evaluate log ( 1 3 ) (c) Evaluate log 1 1. Find the domain of the logarithmic function f() = log ( 1). 3. Find the domain of f() = log(9 + ).. (a) Change the equation log 1 5 = to eponential form. (b) Change the eponential equation 5 = 5 to logarithmic form. 5. (a) Write the equation log 1 = 3 in eponential form. (b) Write the eponential equation 9 5 = 5909 in logarithmic form.. Consider the functions f() = 5 and g() = log 5 (a) Complete the following table of values for f() = (b) Complete the following table of values for g() = log 5 b filling in the -values for the given -values. log What do ou notice about the table of (a) and (b)? (c) Sketch the graphs of f() = 5 and g() = log 5 on the same coordinate aes. (d) Using our answer from (b), sketch h() = log 5 ( + ) Consider the functions f() = ( 1 5) and g() = log 1 5 (a) Complete the following table of values for f() = ( 1. 5) ( 1 5) (b) Complete the following table of values for g() = log 1 b filling in the -values for 5 the given -values. log What do ou notice about the table of (a) and (b)? Page 9

10 (c) Sketch the graphs of f() = ( 1 5) and g() = log 1 on the same coordinate aes. 5 (d) Using our answer from (b), sketch h() = log 1 ( 1) Consider the function f() = log 7 ( + ) (a) Determine the domain of f. Write our answer in interval notation. (b) Solve the equation = log 7 (+) for b converting the equation to eponential form. (c) Use our answer in (b) to complete the following table of values where has been chosen first (d) With the the help of our table in (c), sketch the graph of f(). 9. Consider the function f() = log 1 ( + ) + 3 (a) Determine the domain of f. Write our answer in interval notation. (b) Solve the equation = log 1 (+)+ for b converting the equation to eponential 3 form. (c) Use our answer in (b) to complete the following table of values where has been chosen first (d) With the the help of our table in (c), sketch the graph of f(). 10. The graph of a function f() = log b is given in both graphs below. (a) (a) Sketch the graph of g() = log b (b) Sketch the graph of h() = log b (c) Use the given graph to find the base b. f (b) f Page 10

11 11. The graph of a function f() = log b is given in both graphs below. (a) f (b) (a) On the same graph, sketch the graph = and the inverse function of f() = log b. (b) On the same graph, sketch the graph of h() = log 1/b (c) Find the inverse function of f() = log b. 1. The common log of the positive number M is given below. (a) Epress M as a power of 10. log(m) = 0. (b) Epress M in scientific notation. Use significant figures in our answer. 13. The common log of the positive numbers M and N are given below. log(m) = and log(n) = (a) Epress M and N as powers of 10. (b) Epress M and N in scientific notation. Use significant figures in our answers. 1. In each item below, describe how the graphs of the two functions given are related. ( ) 1 (a) g() = and h() = (b) f() = log () and g() =. (c) k() = 3 and g() =. (d) l() = ( ) + 3 and g() =. (e) f() = log () and m() = log 1 (). f Page 11

12 Properties of Logarithms ( ) 5 z 1. Epand log 5 b and write in terms of log b, log b and log b z (assume > 0, > 0, and z > 0). ( ) in terms of log 5 z 3 1, log 1,. Use properties of logarithms to full epand log 1 z and number(s); simplif our answer (assume > 0, > 0, and z > 0). log 1 3. Use properties of logarithms to write 3 + log 3 log 7 log z as a single logarithm with coefficient 1.. Write 3 log b ( 5 z 5 ) log b (z ( ) ) + 5 log b as a single logarithm with coefficient Use properties of logs and eponents to evaluate: (a) e 5 ln 3 (b) 1 3 log 1 (c) log ( 10 ). (Richter Scale) The magnitude of an earthquake of intensit I on the Richter scale is ( ) I M = log where I 0 is the intensit of a zero-level earthquake. (a) Find the magnitude to the nearest 0.1 of an earthquake that has an intensit of I = 511I 0. ( ) I (b) Write the formula M = log in eponential form, and then solve for I. I 0 (c) Use the formula in (b) to find the intensit of an earthquake that measures 5. on the Richter scale. (d) How man times more intense is an earthquake with a magnitude of.5 than an earthquake with a magnitude of 5.1? Epress answer to nearest whole number. (To do this, find the intensit of each earthquake using our formula from (b), and then divide to find the ratio of the intensities). 7. The range of sound intensities that the human ear can detect is ver large, so a logarithmic scale is used to measure them. The decibel level (db) of a sound is given b I 0 db(i) = 10 log I I 0 where I 0 is the intensit of sound that is barel audible to the huma ear. (a) How man times as great is the intensit of a sound that measures 153 decibels when compared to a sound that measures 113 decibels? (b) Find the decibel level of a sound whose intensit is 171 times as intense as a sound that measures 113 decibels. Round our answer to one decimal place. Page 1

13 . The ph of a solution with a hdronium-ion concentration [H + ] mole per liter is given b ph = log[h + ]. The higher the concentration of [H + ], the more acidic a solution is considered to be. Therefore, because of the negative sign in the above formula, the lower the ph, the more acidic a solution is considered to be. In particular, a solution with concentration of above 10 7 is considered to be acidic whereas a solution with concentration of below 10 7 is considered to be basic. (a) Therefore, a solution with a ph (above/below) is acidic, and a solution with a ph (above/below) is basic. (b) A sample of blood is found to have a hdronium-ion concentration of about mole per liter. What is its ph? Epress answer to 1 decimal place. (c) A swimming pool s water was tested and the ph level was 7.1. Find the a hdroniumion concentration in mole per liter. Use 3 significant figures in our answer. 9. The percentage of a certain radiation that can penetrate millimeters of lead shielding is given b P () = 100e 1.5 (a) Find the percentage of radiation that can penetrate mm of lead shielding. Use at least 3 significant figures in our answer. (b) How man millimeters of lead shielding are required so that less than percent of radiation will penetrate the shielding? Round answer to one decimal place (nearest tenth of a millimeter). 10. Which is bigger 0 or 0? 11. Logarithms are ideall suited for dealing with large numbers, not onl can ou find, for eample, that 75 7 is bigger than 7 75 b comparing logarithms, it is eas to convert a logarithm to scientific notation which gives (perhaps more meaningful) information on the size of a number. The first step is to remember log() = a means = 10 a and properties of eponents then make it eas to convert the number in scientific notation. For eample Therefore, log(75 7 ) = 7 log(75) 7(.3135) = = = This is a huge number, it has 15 digits, but scientific notation, at least, provides a scale of its magnitude. Use properties of logarithms and eponents to answer the following questions. (a) Write in scientific notation. Use significant figures in our final answer. (b) Suppose log(n) = 3.31, write N in scientific notation. (c) Use properties of logarithms, as in the above eample, to write 9 7 in scientific notation. Use significant figures in our final answer. (Use a calculator to check our answer). Page 13

14 (d) Use properties of logarithms, as in the above eample, to write in scientific notation. Use significant figures in our final answer. (Man calculators cannot process numbers above , so ou ma not be able to use a calculator to check our answer). (e) Find the product of numbers 9 7 and Epress our answer in scientific notation. Use significant figures in our answer. 1. Logarithms are ideall suited for dealing with large numbers. Use properties of logarithms and eponents to answer the following questions. (a) Use properties of logarithms to write 7 17 in scientific notation. Use significant figures in our final answer. (b) Find the product Epress our answer in scientific notation. Use significant figures in our answer. (c) Find the quotient Epress our answer in scientific notation. Use significant figures in our answer. Page 1

15 5 Eponential and Logarithmic Equations 1. (a) Find the eact solution to (3 ) 7(9 ) = 0. (b) Use a calculator to epress our answer in (a) to five decimal places.. Solve the equation 7 = 1 3 for. Leave answer in eact form. 3. (a) Find the eact solution to the equation e 0.e the eact solution. e + e (b) Use a calculator to epress our answer in (a) to si decimal places. = 1. You don t have to simplif 3. Find the eact solution(s) to the equation e + e = 11. Verif that our solutions work. 5. Solve the equation log 5 ( ) = 3.. Alwas be careful to check that our solutions work when solving logarithmic equations, because logarithms and their properties are onl defined for positive numbers. See what happens when ou solve the equation: ln() + ln( ) = ln( ) For this, combine the logs on the left side using: ln(m) + ln(n) = ln(mn) when M > 0 and N > 0, and then use the propert ln(e) = ln(f ) implies E = F when ln(e) and ln(f ) are defined. 7. Solve the equation log 5 ( + ) + log 5 ( + 7) = log 5 (5 + ).. The population of a cit is currentl and is epected to grow at a rate of 5.0 percent per ear for the foreseeable future. Its population is given b P (t) = 00000(1.050 t ) where t is the number of ears from toda. (a) What will the population be in 3 ears? (Epress answer as a whole number) (b) At this rate of growth, how long (in ears) will it take the population to double? How long (in ears) would it take the population to quadruple? Epress answers to 1 decimal place. (c) If this growth rate could continue, how long (in ears) would it take for the population to reach 3,000,000 people? Epress answer to 1 decimal place. Page 15

16 Applications of Eponentials and Logarithms 1. (Interest Income) Use the properties of logarithms and eponentials, along with the compound interest formula ( A = P 1 + r ) nt n to answer the following questions. (a) Suppose $1000 is invested at an annual interest rate of.0% compounded monthl. How much will it be worth after 5 ears? Epress answer to nearest penn. (b) How long will it take until the investment is worth $110000? Epress our answer in ears rounded to one decimal place.. (Eponential Growth) The population of bacteria in a vat of potato salad at Bob s All Da Buffet is modeled b P (t) = P 0 e kt. At noon there were 1050 bacteria present and at 1:00 pm there were 1750 bacteria present. (a) Find the specific model for P (t) (i.e., use the information given to find P 0 and k, and plug those values into P (t) = P 0 e kt ), and then use it to answer the following questions. (b) How ma bacteria were present in the potato salad at 11:30 am when was placed in the buffet? Epress answer to nearest whole number. (c) How ma bacteria were present in the potato salad at 3:00 pm when the potato salad was removed from the buffet? (d) If the potato salad had been allowed to remain in the buffet indefinitel, and the model for the bacteria remained valid, how man hours after noon, when there were 1050 bacteria present would it have taken for the bacteria population to reach 915. Epress answer in hours, rounded to nearest decimal place. (For all of these questions, assume no one ate took an potato salad because of its funn smell and hence the population of bacteria remained in tact and followed the given growth model). 3. The population of a small cit is currentl and is growing at percent per ear. Thus the population is given b P (t) = 59000(1.0) t where t is time measured in ears from the present. (a) What will the population of the cit be in one ear? (b) According to this model, what will the population of the cit be in 1 ears from now? Epress answer to the nearest whole number. (c) If this rate of growth continues, how long will it take for the population to triple. Epress answer in ears rounded to two decimal places.. A population of a town follows an eponential growth model P (t) = Ae kt where t is measured in ears, and P (t) is the population at time t. Suppose the population was eactl 7 ears ago, and it is toda. (a) Use this information, with t = 0 for the time eactl 7 ears ago, to find P (t). (b) According to this model, what will the population be 7 ears from now (epress answer to nearest whole number)? Page 1

17 5. (Carbon Dating) Carbon-1 has a half-life of 5730 ears, and satisfies the eponentialdeca equation N(t) = N 0 ( 1 ) t/5730. (a) If an ancient scroll is discovered to have 5.0% of its original Carbon-1, how old is the scroll? Round answer to nearest ear. (b) What percentage of a bone s original Carbon-1 would ou epect to find remaining in a bone that is 00 ears old? Round to the nearest tenth of one percent.. An unknown radioactive element decas into non-radioactive substances. In 50 das the radioactivit of a sample decreases b 5 percent, that is percent of the original substance remains. (a) What is the half-life of the element? (b) How long will it take for a sample of 100 mg to deca to 0 mg? 7. According to Newton s Law of Cooling an object placed in a refigerator with a constant temperature of 39 F has its temperature (in degrees Fahrenheit) given b T (t) = 39 + Ce kt where C and k are constants. Suppose a can of soda (the soda is rather warm because the can was in a warm car) had a temperature of 10 F when it was placed in the refrigerator and 0 minutes later, the soda has cooled to F. (a) Find the C and k for the temperature equation above. Use t in minutes with t = 0 being the time when the can of soda was placed in the refrigerator. (b) What will be the temperature of the can of soda after 5 minutes? Epress answer to the nearest degree. (c) How long will it take for the can of soda to reach 5 F. Epress answer to the nearest minute. A cup containing hot tea is placed in a room whose temperature is kept at a constant 7 F. The tea cooled from 05 F to 15 F in 10 minutes. Use Newton s law of cooling to determine how man more minutes, after the temperature reached 15 F, it will take for the tea to cool to 15 F. Epress answer to 3 decimal places. Note: Newton s law of cooling implies T (t) = 7 + Ce kt where t is measured in minutes. Page 17

Math 121. Practice Problems from Chapter 4 Fall 2016

Math 121. Practice Problems from Chapter 4 Fall 2016 Math 11. Practice Problems from Chapter Fall 01 1 Inverse Functions 1. The graph of a function f is given below. On same graph sketch the inverse function of f; notice that f goes through the points (0,