Math M: Lecture Notes For Chapter 0 Sections 0.: Inverse Function Inverse function (interchange and y): Find the equation of the inverses for: y = + 5 ; y = + 4 3 Function (from section 3.5): (Vertical line crosses only once) profit profit year year A B C Graph A: A function, vertical line crosses only once. Different input, Different output. Graph B: A function, vertical line crosses only once. Different input, Same output (the profit in two different years were the same). Graph C: Not a function, vertical line crosses more than once. Same input, Different output (two different profits for the same year). One -to-one Function: (Horizontal line crosses only once, Different input, Different output) D E F Graph D: A one-to-one, horizontal line crosses only once. Different input, Different output. Graph E: Not one-to-one function, horizontal line crosses more than once. Different input, Same output. Graph F: A one-to-one, horizontal line crosses only once. Different input, Different output. Note: If the function is not one -to-one, then the inverse is not a function. (if you graph the inverse of E which is not one-to-one, you get graph C which is not a function)
Eample : Find the inverse of f ( ) =. Solution: Rewrite it as y =. Replace by y and y by : = y + Isolate y: y =, then f If the function is one-to-one, find its inverse:. ( ) = + a) f () = {(,3), (,4),(3,8), (4,3)} b) f () = {(,), (,5),(3,7)} c) f () = 5 - d) f () = 4 + e) f () = 4 f) f () = 3 + 4 g) f () = + 4 ; ( 4 h) f () = 9 i) f () = +, you cannot have negative under the square root) The following is one-to-one. Graph the original function and its inverse: f ( ) =, ( )
Sections 0.: Eponential Function f() = a (a > 0 and a ) a) Graph: y = 3 b) Graph: y = 4 Note : When a > (as in graph a where a = 3) : The graph rises from left to right When a < (as in graph b where a = 0.5) : The graph falls from left to right Solve the following equations: a) 5 = 5 b) 8 = / c) 4-3 = 64 d) 0 = 0. 4 e) 3 = 7 64 f) 9 8 = 4 7 3
Sections 0.3: Logarithmic Functions y = log a a =. Convert the following from eponential form to logarithmic form: a) y= Answer: log y= b) 3 = 8 Answer: log 8= 3 c) 9 / = 3 Answer: log 9 3= / d) 0 = 00 Answer: log 0 00=. Convert the following from logarithmic to eponential form: a) log 9= 3 Answer: 3 = 9 b) log 5 5= Answer: 5 = 5 c) y = log 4 Answer: y = 4 d) log a = y Answer: a y = 3. Solve for : a) log = 5 Answer: = 5 =3 b) log 5 = 0 Answer: = c) log = -3 Answer: = /8 d) log 7 = /3 Answer: = 3 More eamples: e) log = f) log = 6 g) log 8 7 = h) log 3 ( + 7) = 4 i) log 4 64 = y Note : Fill the missing part: a) log 5... = Answer: log 5 5 = b) log 0... = Answer: log 0 0 = c) log 3... = Answer: log 3 3 = log a a = when the two parts are equal, the answer is Note : Fill the missing part: a) log 5... = 0 Answer: log 5 = 0 b) log 0... = 0 Answer: log 0 = 0 c) log 3... = 0 Answer: log 3 = 0 log = 0 a the log of is always = 0 4
4. Find: a) log 0 0 b) log 0 00 c) log 5 5 d) log 0 000 5. Graph the following: y = log (same as graphing: y = ) The following is problem 56 from the book, page 63: A study showed that the number of mice in an old abandoned house was approimated by the function defined by M ( t) = 6 log 4 (t + 4) where t is measured in months and t = 0 corresponds to January 998. Find the number of mice in the house in (a) January 998 (b) July 998 (c) July 000 (d) Graph the function. 5
Sections 0.4: Properties of Logarithmic Functions Properties of logarithms Rule Formula Eamples I) Multiplication Rule : log of multiplication = sum of log II) Division Rule : log of division = difference of log log(.y) = log + log y log y log 5 = log5 + log = log log y log 5/ = log 5 - log III) Power Rule log r = r. log log 5 = log 5 Important: The following are WRONG: log. log y = log + log y (compare it to rule I) log = log log y (compare it to rule II) log y (log ) = r. log (compare it to rule III) r. Epress as a sum or difference of logarithms: a) log( y 3 y ) b) log 4 z c) log 3 y 3 5 z w. Epress as a single logarithm: a) 3 log / log y + 3 log z b) log - 3 log y - log z c) log ( - 9) - log ( + 3) d) log - log 3. Given log b =. and log b 3 =.. Find a) log b 6 b) log b c) log b b d) log b 3 e) log b 9 f) log b 6
Sections 0.5: Common and Natural Logarithms y = log a a = y The number a is called the logarithmic base If a = 0, then it is log 0 and it is called Common logarithm (available in calculator as log) If a = e, then it is log e or ln and it is called Natural logarithm (available in calculator as ln) log 0 = log ( Common Logarithm) log e = ln ( Natural Logarithm) log 5 is the same as log 0 5 ; ln 5 is the same as log e 5 Reminder: log a a = or log e e = ln e = log a = 0 or log e = 0 ln = 0 Changing logarithm Base: log log a = log Use the change-of-base rule to find each logarithm (round the answer to 4 decimal places) b b a a) log 5 8 b) log 3 7 c) log 0.08 d) log π e 7
a) Graph: y = e and y = e y = e y = e 0 - b) Graph: y = ln y = ln 0 3 compare this graph with y = e c) Graph: y = ln 3 y = ln 3 0 4 50 What happens between = 4 and when =50? 8
Sections 0.6: Applications. Solve without using calculator: a) log = -4 b) log 9 = / c) ln = -3 d) 5 = 0 e) 3 = 0 f) ln = 4. Solve using calculator and give solutions to three decimal places: + a) 6 = b) 5 3 = c) + 6 3 = 4 3. Solve using the natural logarithm and give solutions to three decimal places: Remember: ln = 0 and ln e = 0.03 0.04 a) e = 7 b) ln e = 3 4. Solve each equation: a) log 4 (3- ) = b) log (- ) = 3 c) log 4 (- 3) 3 = 4 5. Solve each equation: a) log( 5 ) = log b) log 4 log( + ) = log c) log 3 log( ) = log d) log + log( 3) = e) log 3 ( ) log 3 ( 4) = f) log 3 ( 4) + log 3( + 4) = g) log + log ( ) = 3 h) log 4 ( + 6) log 4 = i) ln( t ) = 3 j) ln( t + ) = 0 9
Compound Interest: A = P. + P: the principal, amount invested A: the new balance t: the time in years r: the rate, (in decimal form) n: the number of times it is compounded. r n nt Eample : Suppose that $5000 is deposited in a saving account at the rate of 6% per year. Find the total amount on deposit at the end of 4 years if the interest is: P =$5000, r = 6%, t = 4 years a) compounded annually, n = : A = 5000( + 0.06/) ()(4) = 5000(.06) (4) = $63.38 b) compounded semiannually, n =: A = 5000( + 0.06/) ()(4) = 5000(.03) (8) = $6333.85 c) compounded quarterly, n = 4: A = 5000( + 0.06/4) (4)(4) = 5000(.05) (6) = $6344.93 d) compounded monthly, n =: A = 5000( + 0.06/) ()(4) = 5000(.005) (48) = $635.44 e) compounded daily, n =365: A = 5000( + 0.06/365) (365)(4) = 5000(.0006) (460) = 6356. Continuous Compound Interest: Continuous compounding means compound every instant where: rt A = P.e and Doubling Time = ln r Eample : Solve the previous eample but with interest compounded continuously. P =$5000, r = 6%, t = 4 years A= 5000.e (0.06)(4) = 5000.(.75) = $6356.4 Eample 3: Find the amount to be invested at a rate of 8% compounded continuously in order to get $,930 in 6 years. A = $,930, r = 0.08, t = 6 years. $,930 = P.e (0.08)(6), then P = $8000 Eample 4: a) what will be the amount A in an account with initial principal of $00,000 if interest is compounded continuously at an annual rate of 5% for 5 years? b) How long will it take for the initial amount to double? Eample 5: How much money must be deposited today to amount to $4000 in 0 years at a rate of 4% compounded continuously? Eample 6: How long does it take an amount to double at a rate of 8.5% compounded continuously? 0