Section 0.4 Inverse functions and logarithms
|
|
- Myra Stone
- 6 years ago
- Views:
Transcription
1 Section 0.4 Inverse functions and logarithms (5/3/07) Overview: Some applications require not onl a function that converts a numer into a numer, ut also its inverse, which converts ack into. In this section we analze inverse functions and discuss logarithms, which are the inverses of eponential functions. Topics: Inverse functions Changing variales Restricting domains Logarithms The common and natural logarithms Laws of logarithms Inverse functions The function of Figure converts the ear t into the numer N = f(t) (millions) of cars that were registered in California at that time. () For eample, starting with t = 980 on the horizontal t-ais, moving verticall to the graph, and then horizontall to the vertical N-ais, we otain N = f(980) = 7.4. At the eginning of 980 there were 7.4 million cars registered in California. 30 N (millions) N = f(t) 000 t (ear) t = f (N) t N (millions) FIGURE FIGURE The inverse of f, denoted f and read f inverse, has the opposite effect. It converts the numer N into the time t = f (N) when there were N cars registered. We could use Figure to stud f having its variale (the independent variale) e on the vertical N-ais and its values on the horizontal t-ais. For eample, we could start with N = 7.4 on the vertical ais, move horizontall to the graph, and then verticall to the horizontal t ais to otain t = 980. This gives f (7.4) = 980, which means that numer of cars registered was 7.4 million at the eginning of 980. it is used. () Data adapted from CALPERG Citizen Agenda, Vol. 8, No., Los Angeles: CALPERG, 000, p. 5. Because the smol f also denotes the reciprocal /f of f, its meaning must e determined from the contet in which
2 p. (5/3/07) Section 0.4, Inverse functions and logarithms This procedure for studing the inverse is, however, not convenient ecause we prefer to have the independent variale on the horizontal ais. To achieve this, we flip the drawing aout the diagonal line that makes a 45 -degree angle with the positive - and -aes. This ields the graph t = f (N) of the inverse function in Figure, where the variale N is on the horizontal ais and the values t of the function are on the vertical ais. The graph of f in Figure is the mirror image of the graph of f in Figure with respect to the diagonal line. Here is a general definition: Definition Suppose that = f() is a function such that for each in its range, the equation = f() has one and onl one solution. Then is the value of the inverse function = f () at. Thus, = f () = f(). () Question Suppose that f has an inverse f, that f() = 0, and that f (30) =. What are the values of f (0) and f()? The domain of f is the range of f and the range of f is the domain of f. Figures 3 and 4 illustrate how the domains and ranges of a function and its inverse are related. The domain of = f() in Figure 3 is the interval a on the horizontal -ais and its range is the interval c d on the vertical -ais. The domain of = f () in Figure 4 is the interval c d on the horizontal -ais and its range is the interval a on the vertical -ais. = f() = f () d c a a c d FIGURE 3 FIGURE 4 The function f and its inverse f undo each other in the sense that f (f()) = for in the domain of f () f ( f () ) = for in the domain of f. (3) Eample Figure 5 shows the graph = h() of the function h() = 3. Find a formula for the inverse function = h () and draw its graph in a -plane. To find a formula for = h (), we solve = 3 for. We otain first 3 = and then = 6 3. Thus, the inverse function = h () is given h () = 6 3. Its graph in the -plane of Figure 6 is the line of slope 3 with -intercept 6 and is the mirror image aout the diagonal line of the graph of = h() in Figure 5. (The diagonal line that makes equal angles with the positive aes in Figures 5 and 6 is = ecause the scales on the aes are equal.) The smol = means implies, = means is implied, and means if and onl if.
3 Section 0.4, Inverse functions and logarithms p. 3 (5/3/07) = = = h() 4 = h () h() = 3 h () = 6 3 FIGURE 5 FIGURE 6 Changing variales In Definition and Eample, the variale of the inverse function = f () was the letter used for values of the original function = f().in some cases it is convenient to have e the variale of oth the function and its inverse. A formula for the inverse function can then often e found the following rule. Rule To find a formula with variale for the inverse of = f(), first solve the equation = f() to otain the inverse = f () with variale. Then interchange the letters and to otain a formula for = f (). The procedure descried in Rule is illustrated in Figures 7 through 9. We egin with the function = f() in the -plane of Figure 7. We solve for to otain the inverse function = f () with variale in the -plane of Figure 8. Then we interchange the letters and to otain the function = f () with variale in the -plane of Figure 9. = f() = f () = f () d c a a a c d c d FIGURE 7 FIGURE 8 FIGURE 9 Question How can Figure 9 e otained directl from Figure 7? Notice that interchanging and to otain = f () from = f () is not the same as solving the latter equation for. That would take us ack to the original equation = f(). When we interchange and, we are not solving an equation, we are just interchanging the letters used for the variale and for values of the function.
4 p. 4 (5/3/07) Section 0.4, Inverse functions and logarithms Eample Figure 0 shows the graph of the function = 3 with the diagonal line that makes equal angles with the positive aes. Find a formula for the inverse of = 3 with as variale and draw its graph. = 3 = 3 = 3 FIGURE 0 FIGURE FIGURE We solve = 3 taking cue roots of oth sides to find the inverse = 3 with as variale. The graph of this function in the -plane of Figure is otained flipping all of Figure 0 aout the dashed line in that drawing. We interchange the letters and to otain = 3. The graph of this function in the -plane of Figure can e otained interchanging and in Figure or reflecting the curve in Figure 0 aout the diagonal line without moving the aes. The net two eamples illustrate the fact that a function and its inverse undo each other if one is applied immediatel after the other. Eample 3 Epress the solutions of 3h( ) = in terms of the function h. We want to use the fact that h (h()) =, so we first divide oth sides of 3h( ) = 3 to otain h( ) = 4. Then, when we appl the function = h () to oth sides, we otain h ( h( ) ) = h (4), which simplifies to = h (4). Finall, taking square roots of oth sides gives the solutions = ± h (4). Eample 4 Epress the solution of S ( + ) = 5 in terms of the function S. It would do us no good to appl the function S to the given equation ecause of the square root. Instead, we first square oth sides to otain S ( + ) = 5. Appling S to oth sides of this equation then gives S ( S ( + ) ) = S(5), which simplifies to + = S(5). We sutract from oth sides to otain = S(5). Restricting domains If a horizontal line = c intersects the graph of a function = f() at more than one point, then the function does not have an inverse, ecause the inverse function cannot have more than one value at = c. The squaring function = in Figure 3, for eample, does not have an inverse ecause the horizontal line = c for an positive c intersects the graph at two points.
5 Section 0.4, Inverse functions and logarithms p. 5 (5/3/07) = = F() = c F() = for 0 FIGURE 3 FIGURE 4 To otain an inverse, we define F to e the function with values = ut defined onl for 0 (Figure 4). Then the horizontal line = c for an c 0 intersects the graph at onl one point. The inverse of F with variale is the square-root function = in Figure 5 and its inverse with variale is = in Figure 6. = = FIGURE 5 FIGURE 6 Eample 5 (a) Eplain wh the function = 3 4 of Figure 7 does not have an inverse. () Draw the graph of the function = G() defined G() = 3 4 for 0. (c) Find a formula for the inverse = G () and draw its graph. = = G() = G () 3 G() = 3 4, 0 G () = 4 3 FIGURE 7 FIGURE 8 FIGURE 9
6 p. 6 (5/3/07) Section 0.4, Inverse functions and logarithms Question 3 (a) The function = 3 4 of Figure 7 does not have an inverse ecause the horizontal line at an < 3 intersects the graph at two points. () The graph of = G() in Figure 8 is the portion of = 3 4 for 0. (c) The equation = G() is equivalent to = 3 4 with 0. To solve for, we sutract 3 from oth sides of the equation to have 3 = 4 with 0. Multipling oth sides and taking fourth roots gives = ± 4 3 and then = 4 3 since 0. Therefore, G () = 4 3, and interchanging and ields G () = 4 3. The graph of = G () in Figure 9 can e otained reflecting the graph of G in Figure 8 aout the diagonal line at 45 to the positive -and -aes. Figure 0 shows the graph of = H() defined H() = 3 4 for 0. Find a formula for = H () and draw its graph. = H() 3 FIGURE 0 H() = 3 4, 0 Logarithms For an constant >, the eponential function = of Figure has an inverse, which is called the logarithm to the ase and is denoted = log (Figure ). Thus, = if and onl if = log. (4) Here is the variale of the eponential function and and is the variale of the logarithm. = = = log = log FIGURE FIGURE FIGURE 3 Interchanging and in Figure gives the graph = log in Figure 3 of the logarithm with variale. It is the mirror image of = aout the diagonal line in Figure. Because = is defined for all and its range is the interval > 0, = log is defined for > 0 and its range is the set of all numers. The fact that = and = log are inverse functions is also epressed in the two equations, log ( ) = for all (5) log = for all > 0. (6)
7 Section 0.4, Inverse functions and logarithms p. 7 (5/3/07) Notice that, while these formulas are visuall quite comple, all the state is that = and = log undo each other. The asic properties of log follow from properties of discussed in the last section. For an >, an n, and an positive and, log () = 0 (7) log () = (8) log () = log + log (9) ) log ( = log log (0) log ( n ) = nlog. () Also, logarithms with different ases, > and c >, are related the formula log = [log c][log c ] = log c log c. () To rememer formulas (), notice that the is higher the in all three epressions and that one c is higher than the other in each of the last two epresssions. Eample 6 Solve = 5 for. = 5 for = log (5). Question 4 Solve log 0 = for. The common logarithm The logarithm to the ase 0, = log 0, is called the common logarithm and is often denoted = log with no suscript. It was emploed to simplif calculations involving products and powers efore there were calculators and computers, and is still used in some applications. Eample 7 In chemistr the ph of a solution is defined ph = log 0 [H + ], where [H + ] is the concentration of hdrogen ions in the solution, measured in moles per liter. ( mole = molecules.) The hdrogen ion concentration of water is 0 7 moles per liter. What is the ph of water? The ph of water is log 0 (0 7 ) = ( 7) = 7. The natural logarithm The logarithm that is most convenient for calculus is = log e, with ase equal to the ase e of the natural eponential function = e that was introduced in the last section. This logarithm is called the natural logarithm and is usuall denoted = ln (read ell n ): Since = e and = ln are inverse functions, ln = log e for all > 0. (3) e ln = for all > 0 (4) ln(e ) = for all. (5) The graphs = e and = ln in Figure 4 are mirror images of each other aout the line = ecause there are equal scales on the aes in that drawing. Eample 8 Solve the equation e 3 = 00 for. C Question 5 We take the natural logarithm of oth sides of the equation to otain ln(e 3 ) = ln(00). Since the natural logarithm and eponential function are inverses, this gives 3 = ln(00). Finall, we divide 3 to otain the solution = 3 ln(00). Check the result of Eample 8 using a calculator or computer to find the approimate decimal value of the solution and then to evaluate e 3 with this numer.
8 p. 8 (5/3/07) Section 0.4, Inverse functions and logarithms = e = FIGURE 4 = ln Eample 9 Find all solutions of (ln ) =. We cannot use the fact that = ln and = e are inverses appling the eponential function to oth sides of the given equation ecause the logarithm is squared. Instead, we first take the square roots of oth sides to have ln = ±. Then appling the eponential function gives the solutions = e = e and = e = /e. C Question 6 Generate the cuve = (ln ) with the line = in the window 5, to illustrate the results of Eample 9. The natural logarithm can e used to find values of logarithms to an ase > using () with c = e: log = ln for all > 0. (6) ln To rememer this formula, notice that the is higher than the on oth sides of the equation. C Eample 0 Give the approimate decimal value of log (3). Formula (6) with = and = 3 gives log (3) = ln(3) ln(). = Responses 0.4 Response f (0) = ecause f() = 0 f() = 30 ecause f (30) = Response To otain the graph of = f () in Figure 9, reflect the graph of = f() in Figure 7 aout the diagonal line =, without moving the - and -aes. Response 3 Solve = 3 4 for with 0. 4 = 3 = 4 3 H () = 4 3 H () = 4 3 The graph of = H () in Figure R3 is the mirror image aout the dashed line of the graph of = H() in Figure 0. = H ().5 = (ln ) = e e 4 Figure R3 Figure R6
9 Section 0.4, Inverse functions and logarithms p. 9 (5/3/07) Response 4 log 0 = for = 0 = 0 Response 5 = 3 ln(00). = e 3( ). = Response 6 Figure R6 shows the curve = (ln ), the line =, and their intersections, which are at = e. =.7 and = e. = Interactive Eamples 0.4 Interactive solutions are on the we page http// ashenk/.. Find approimate values of (a) G(4), () G (0, 000), and (c) G (G(3) )), where G is the function whose graph is in Figure = G() FIGURE Solve e 4 = 000 for. Then generate = e 4 and = 000 in the window.5.5, and use an intersect or trace operation to check our answer. 3. Solve + = + for. Then generate = + and = + in the window, 7, and use an intersect or trace operation to check our answer. 4. Solve ln( 5 ) = for. 5. Solve 4 = 5(8 ) for. 6. Solve [ln(5)] = 9 for 7. Solve F (F(F ())) = under the assumption that = F() and its inverse = F () are defined for all. Eercises 0.4 A Answer provided. CONCEPTS: O Outline of solution provided. C Graphing calculator or computer required.. Simplif the epresions (a) f ( f (f()) ) and () f ( f ( f () )), where is a numer in the domains of f and f.. (a) What are the values of log 0 (00), log 0 (000), and log 0 (00,000)? () How are the numers from part (a) related and what propert of logarithms is illustrated this relationship? 3. (a) What are the values of ln(e 3 ), ln(e 5 ), and ln(e 3 e 5 )? () How are the numers from part (a) related and what propert of logarithms is illustrated this relationship? 4. (a) What are the values of 3 ln(e 5 ), and ln ( (e 5 ) 3)? () How are the numers from part (a) related and what propert of logarithms is illustrated this relationship? In the pulished tet the interactive solutions of these eamples will e on an accompaning CD disk which can e run an computer rowser without using an internet connection.
10 p. 0 (5/3/07) Section 0.4, Inverse functions and logarithms BASICS: 5. Epress the solutions of (a) f() = 4 and () f () = 7 in terms of values of f and f. 6. O What are (a) P(), () P (0.), and (c) P ( P (0.) + ) if P is given the following tale? P() O (a) Find a formula for = f () where f() = /3. C () Generate the graphs of = f() and = f () with the line = in an -plane and cop them on our paper. Use a window with equal scales on the aes that includes the square,. 8. O Solve (a) log 0 ( ) = 4, () 0 = 5, (c) ln =, and (d) e = 7 for. Give eact answers and approimate decimal values. C Check our answers with a calculator or computer. 9. O Use properties of the common logarithm to show that each of the numers (a) (c) in the first column elow equals one of the numers (I) (III) in the second column. (a) log 0 (300) (I) log 0 (3) () log 0 (30) log 0 (3) (II) + log 0 (3) (c) log0 (8) (III) 0. O Figure 6 shows the graph of a function = H(), defined for 0 6. Draw the graph of its inverse = H (). 6 4 = H() FIGURE The graph of = k() is in Figure 7. Draw the graph of its inverse = k (). = k() 4 FIGURE 7 4
11 Section 0.4, Inverse functions and logarithms p. (5/3/07). O The function r = G(v) of Figure 8 gives the rate of gasoline consumption of a car, measured in miles per gallon, as a function of the car s velocit v, measured in miles per hour. (a) What is the action of the inverse function v = G (r)? () What is approimate value of G (5)? 0 r (miles per gallon) r = G(v) 5 0 FIGURE v (miles per hour) 3. O Figure 9 shows the graph of the function P = F(Q) that converts the numer of quarts Q in a volume into the numer of points P. (a) What is the action of Q = F (P)? () Give formulas for P = F(Q) and Q = F (P). (c) Draw the graph of Q = F (P) in a PQ-plane P (pints) P = F(Q) 6 4 FIGURE Q (quarts) 4. A An airplane that is fling west at the constant speed of 500 miles per hour flies s = f(t) miles in t (hours) for t 0, where f(t) = 500t. (a) What is the action of t = f (s)? () Find a formula for t = f (s). Then draw the graph s = f(t) in a ts-plane and the graph of t = f (s) in an st-plane. 5. A grain silo contains 00 tons of wheat at the eginning of an eight-hour work da. During the da wheat is removed at the constant rate of 0 tons per hour, so that the weight of wheat in the silo t hours after the eginning of the work da is w = F(t) tons, where F(t) = 00 0t of (a) What does F (00) represent in terms of weight and time? () Give a formula for t = F (w). 6. A The volume of a sphere of radius r is V = G(r) with G(r) = 4 3 πr3 for r 0. Give a formula for r = G (V ). What does this function do? 7. If V gallons of a salt solution contains 5 pounds of salt, then the concentration of the solution is = f(v ) pounds per gallon, where f(v ) = 5/V. (a) What does the inverse function V = f () do? () Find a formula for V = f (). (c) Draw the graph of = f(v ) in a V -plane and the graph of V = f () in a V -plane. Use V 8 and 8 and have equal scales on the aes in oth drawings. 8. O (a) Find a formula for the inverse of = 3 + as a function of. C () Generate the graphs of oth functions with = in a window that includes 3 3, 3 3 and has equal scales on the aes. Cop the drawing on our paper. If our calculator or computer does not generate the graph of 3 + for <, enter it as the two functions (+) (/3) for and (as( + )) (/3) for <. Use the setting ZSquare under Zoom to get equal scales on a TI calculator.
12 p. (5/3/07) Section 0.4, Inverse functions and logarithms 9. O (a) Find a formula for the inverse of = / as a function of. () Draw the graphs of oth functions in separate -planes with 4 6, 4 6 and equal scales on the aes. 0. (a) Find a formula for the inverse of = 3 as a function of. C () Generate the graphs of oth functions with = in a window that has equal scales on the aes and that includes the square.5.5,.5.5. Cop the drawing on our paper. In Eercises through 3, (a) solve the required equations for. C () Then generate the curves in the given windows and use an intersect or trace operation or the curves as a partial check of our answers.. Solve ln(00 + ) = 6. (Generate = ln(00 + ) and = 6 in the window , 8 with -scale = 00.). Solve = 0. (Generate = and = 0 in the window.5.5, ) 3. Solve 50 (0 ) = 50. (Generate = (0 ) and = 00 in the window, with -scale = 50.) Solve the equations in Eercises 4 through 30 for. 4. A 3 = 7 8. O e = = 5 9. ln( ) + ln( ) = 5 6. = ln(e ) = 3 7. A 4 8 = 0 The functions in Eercises 3 through 33 and their inverses are defined for all. Solve the equations for. 3. A 3A() = f() = 4 f() EXPLORATION: 33. P(P ()) = A The 00 California state ta T (dollars) for a single person on taale income of I dollars with 0 I 3,65 is given T = f(i) with () f(i) = { 0.0I for 0 I (I 5748) for 5459 I,939. (a) Sketch the graph T = f(i) in an IT-plane. () Descrie in terms of income and taes the action of the function f. (c) Find a formula similar to (7) for I = f (T) and sketch its graph in a TI-plane. C A o that weighs 00 pounds on the surface of the earth weighs w = ( pounds when h) it is h miles aove the surface of the earth. (a) Generate the graph of this function on our calculator or computer, using 000 h 0, 000, h-scale = 000, 0 w 50, and w-scale = 50, and cop it on our paper. () Give a formula for the height of the o aove the surface of the earth as a function of its weight. Generate the graph of this function in a wh-plane with the same ranges as in part (a), and cop it on our paper. 36. Find a formula for the inverse = h () of = h() defined h() = / for > 0. Then draw the graphs of oth functions with equal scales on the aes. () Data from 00 Personal Income Ta Booklet, Sacramento, CA: Franchise Ta Board, 00, p. 66.
13 Section 0.4, Inverse functions and logarithms p. 3 (5/3/07) In Eercises 37 through 39 (a) solve the equations for. C () Generate the graphs of the functions on oth sides of the equations in sutale windows to check our answers. 37. A ln( + ) = 38. A = 39. = 6( ) 40. The magnitude of an earthquake on the Richter scale is M = 0.67 log 0 (0.37E) +.46, where E is the energ of the earthquake in kilowatt hours. What is the energ (a A ) of an earthquake of magnitude 5 and () of an earthquake of magnitude 6? Give approimate decimal values. 4. The loudness of sound can e measured in deciels, where one deciel if supposed to e the smallest change in loudness that the average human can detect. (One el, named in honor of Aleander Graham Bell equals ten deciels.) Sound reaches the eardrum as an oscillation in the air pressure. If the variation in air presure is P pounds per square inch, then the loudness of the sound is D = 0log 0 [( )P] deciels. What is the variation in sound pressure caused (a) a whisper at 0 deciels and () a rock and at 0 deciels? 4. The magnitude of a star is defined the formula M =.5log 0 (ki), where k is a positive constant and I is the intensit of the light from the star. (a) Do righter stars have less or greater magnitudes than dimmer stars? () What is the ratio of the intensities of light from the rightst star Sirius, which has a magnitude of.6 and from the star Betelgeuse, which has a magnitude of 0.9? In Eercises 43 through 46 solve the equations for under the assumption that the functions and their inverses are defined for all. 43. O 4 G() = G() 44. A H()H () = [R()] R() + = S () S () + = 6 (End of Section 0.4)
ab is shifted horizontally by h units. ab is shifted vertically by k units.
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS
Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic
More information3.3 Logarithmic Functions and Their Graphs
274 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions What ou ll learn about Inverses of Eponential Functions Common Logarithms Base 0 Natural Logarithms Base e Graphs of Logarithmic Functions
More informationSection 1.5 Formal definitions of limits
Section.5 Formal definitions of limits (3/908) Overview: The definitions of the various tpes of limits in previous sections involve phrases such as arbitraril close, sufficientl close, arbitraril large,
More informationExponential, Logistic, and Logarithmic Functions
CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic Functions and Their Graphs 3.4 Properties of Logarithmic
More informationChapter 12 Exponential and Logarithmic Functions
Chapter Eponential and Logarithmic Functions. Check Points. f( ).(.6) f ().(.6) 6.86 6 The average amount spent after three hours at a mall is $6. This overestimates the amount shown in the figure $..
More information7.4. Characteristics of Logarithmic Functions with Base 10 and Base e. INVESTIGATE the Math
7. Characteristics of Logarithmic Functions with Base 1 and Base e YOU WILL NEED graphing technolog EXPLORE Use benchmarks to estimate the solution to this equation: 1 5 1 logarithmic function A function
More information1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs
0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs.5 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals
More informationf 0 ab a b: base f
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential
More information7-1. Exploring Exponential Models. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Cross out the expressions that are NOT powers.
7-1 Eploring Eponential Models Vocabular Review 1. Cross out the epressions that are NOT powers. 16 6a 1 7. Circle the eponents in the epressions below. 5 6 5a z Vocabular Builder eponential deca (noun)
More informationP.4 Lines in the Plane
28 CHAPTER P Prerequisites P.4 Lines in the Plane What ou ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables
More informationNumber Plane Graphs and Coordinate Geometry
Numer Plane Graphs and Coordinate Geometr Now this is m kind of paraola! Chapter Contents :0 The paraola PS, PS, PS Investigation: The graphs of paraolas :0 Paraolas of the form = a + + c PS Fun Spot:
More informationPRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 3: Exponential, Logistic, and Logarithmic Functions 3.1: Exponential and Logistic Functions
PRE-CALCULUS: Finne,Demana,Watts and Kenned Chapter 3: Eponential, Logistic, and Logarithmic Functions 3.1: Eponential and Logistic Functions Which of the following are eponential functions? For those
More informationLogarithmic Functions. 4. f(f -1 (x)) = x and f -1 (f(x)) = x. 5. The graph of f -1 is the reflection of the graph of f about the line y = x.
SECTION. Logarithmic Functions 83 SECTION. Logarithmic Functions Objectives Change from logarithmic to eponential form. Change from eponential to logarithmic form. 3 Evaluate logarithms. 4 Use basic logarithmic
More informationChapter 9 Vocabulary Check
9 CHAPTER 9 Eponential and Logarithmic Functions Find the inverse function of each one-to-one function. See Section 9.. 67. f = + 68. f = - CONCEPT EXTENSIONS The formula = 0 e kt gives the population
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationChapter 4 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 4. to accompany. College Algebra by Julie Miller
Chapter 4 Page 1 of 16 Lecture Guide Math 105 - College Algebra Chapter 4 to accompan College Algebra b Julie Miller Corresponding Lecture Videos can be found at Prepared b Stephen Toner & Nichole DuBal
More informationThe Natural Base e. ( 1, e 1 ) 220 Chapter 3 Exponential and Logarithmic Functions. Example 6 Evaluating the Natural Exponential Function.
0 Chapter Eponential and Logarithmic Functions (, e) f() = e (, e ) (0, ) (, e ) FIGURE.9 The Natural Base e In man applications, the most convenient choice for a base is the irrational number e.78888....
More informationLinear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?
3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear
More information3.2 Logarithmic Functions and Their Graphs
96 Chapter 3 Eponential and Logarithmic Functions 3.2 Logarithmic Functions and Their Graphs Logarithmic Functions In Section.6, you studied the concept of an inverse function. There, you learned that
More informationEvaluate Logarithms and Graph Logarithmic Functions
TEKS 7.4 2A.4.C, 2A..A, 2A..B, 2A..C Before Now Evaluate Logarithms and Graph Logarithmic Functions You evaluated and graphed eponential functions. You will evaluate logarithms and graph logarithmic functions.
More informationPractice A ( 1, 3 ( 0, 1. Match the function with its graph. 3 x. Explain how the graph of g can be obtained from the graph of f. 5 x.
8. Practice A For use with pages 65 7 Match the function with its graph.. f. f.. f 5. f 6. f f Lesson 8. A. B. C. (, 6) (0, ) (, ) (0, ) ( 0, ) (, ) D. E. F. (0, ) (, 6) ( 0, ) (, ) (, ) (0, ) Eplain how
More informationMath 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,
More information5.6. Differential equations
5.6. Differential equations The relationship between cause and effect in phsical phenomena can often be formulated using differential equations which describe how a phsical measure () and its derivative
More informationMath 121. Practice Problems from Chapter 4 Fall 2016
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
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions.1 Eponential Growth and Deca Functions. The Natural Base e.3 Logarithms and Logarithmic Functions. Transformations of Eponential and Logarithmic Functions.5 Properties
More informationSTRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula
UNIT F4 Solving Quadratic Equations: Tet STRAND F: ALGEBRA Unit F4 Solving Quadratic Equations Tet Contents * * Section F4. Factorisation F4. Using the Formula F4. Completing the Square UNIT F4 Solving
More informationUNIT TWO EXPONENTS AND LOGARITHMS MATH 621B 20 HOURS
UNIT TWO EXPONENTS AND LOGARITHMS MATH 61B 0 HOURS Revised Apr 9, 0 9 SCO: By the end of grade 1, students will be epected to: B30 understand and use zero, negative and fractional eponents Elaborations
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
More informationThe formulas below will be provided in the examination booklet. Compound Interest: r n. Continuously: n times per year: 1
HONORS ALGEBRA B Semester Eam Review The semester B eamination for Honors Algebra will consist of two parts. Part will be selected response on which a calculator will not be allowe Part will be short answer
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More information1.7 Inverse Functions
71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,
More informationSTRAND: ALGEBRA Unit 2 Solving Quadratic Equations
CMM Suject Support Strand: ALGEBRA Unit Solving Quadratic Equations: Tet STRAND: ALGEBRA Unit Solving Quadratic Equations TEXT Contents Section. Factorisation. Using the Formula. Completing the Square
More informationExponential and Logarithmic Functions
7 Eponential and Logarithmic Functions 7.1 Eponential Growth and Deca Functions 7. The Natural Base e 7.3 Logarithms and Logarithmic Functions 7. Transformations of Eponential and Logarithmic Functions
More information1. For each of the following, state the domain and range and whether the given relation defines a function. b)
Eam Review Unit 0:. For each of the following, state the domain and range and whether the given relation defines a function. (,),(,),(,),(5,) a) { }. For each of the following, sketch the relation and
More informationExponential and Logarithmic Functions
7 Eponential and Logarithmic Functions In this chapter ou will stud two tpes of nonalgebraic functions eponential functions and logarithmic functions. Eponential and logarithmic functions are widel used
More informationExponentials and Logarithms Review Part 2: Exponentials
Eponentials and Logaritms Review Part : Eponentials Notice te difference etween te functions: g( ) and f ( ) In te function g( ), te variale is te ase and te eponent is a constant. Tis is called a power
More information13. x 2 = x 2 = x 2 = x 2 = x 3 = x 3 = x 4 = x 4 = x 5 = x 5 =
Section 8. Eponents and Roots 76 8. Eercises In Eercises -, compute the eact value... 4. (/) 4. (/). 6 6. 4 7. (/) 8. (/) 9. 7 0. (/) 4. (/6). In Eercises -4, perform each of the following tasks for the
More informationThe semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. n times per year: 1
ALGEBRA B Semester Eam Review The semester B eamination for Algebra will consist of two parts. Part 1 will be selected response. Part will be short answer. Students ma use a calculator. If a calculator
More informationName Date. Logarithms and Logarithmic Functions For use with Exploration 3.3
3.3 Logarithms and Logarithmic Functions For use with Eploration 3.3 Essential Question What are some of the characteristics of the graph of a logarithmic function? Every eponential function of the form
More information9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson
Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric
More informationSTANDARD FORM is a QUADRATIC FUNCTION and its graph is a PARABOLA. The domain of a quadratic function is the set of all real numbers.
EXERCISE 2-3 Things to remember: 1. QUADRATIC FUNCTION If a, b, and c are real numbers with a 0, then the function f() = a 2 + b + c STANDARD FORM is a QUADRATIC FUNCTION and its graph is a PARABOLA. The
More information2.2 Equations of Lines
660_ch0pp07668.qd 10/16/08 4:1 PM Page 96 96 CHAPTER Linear Functions and Equations. Equations of Lines Write the point-slope and slope-intercept forms Find the intercepts of a line Write equations for
More information5.1 Exponential and Logarithmic Functions
Math 0 Student Notes. Eponential and Logarithmic Functions Eponential Function: the equation f() = > 0, defines an eponential function for each different constant, called the ase. The independent variale
More informationUse Properties of Exponents
4. Georgia Performance Standard(s) MMAa Your Notes Use Properties of Eponents Goal p Simplif epressions involving powers. VOCABULARY Scientific notation PROPERTIES OF EXPONENTS Let a and b be real numbers
More informationExponents and Exponential Functions
Chapter Eponents and Eponential Functions Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. Prerequisite Skills for the chapter Eponents and Eponential Functions. : eponent, ase.
More informationFunctions. Essential Question What are some of the characteristics of the graph of a logarithmic function?
5. Logarithms and Logarithmic Functions Essential Question What are some o the characteristics o the graph o a logarithmic unction? Ever eponential unction o the orm () = b, where b is a positive real
More informationOBJECTIVE 4 EXPONENTIAL FORM SHAPE OF 5/19/2016. An exponential function is a function of the form. where b > 0 and b 1. Exponential & Log Functions
OBJECTIVE 4 Eponential & Log Functions EXPONENTIAL FORM An eponential function is a function of the form where > 0 and. f ( ) SHAPE OF > increasing 0 < < decreasing PROPERTIES OF THE BASIC EXPONENTIAL
More informationExponential and Logarithmic Functions, Applications, and Models
86 Eponential and Logarithmic Functions, Applications, and Models Eponential Functions In this section we introduce two new tpes of functions The first of these is the eponential function Eponential Function
More informationGraphs and polynomials
1 1A The inomial theorem 1B Polnomials 1C Division of polnomials 1D Linear graphs 1E Quadratic graphs 1F Cuic graphs 1G Quartic graphs Graphs and polnomials AreAS of STud Graphs of polnomial functions
More informationGraphs and polynomials
5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Graphs and polnomials VCEcoverage Areas of stud Units & Functions and graphs Algera In this chapter A The inomial
More informationMath 0210 Common Final Review Questions (2 5 i)(2 5 i )
Math 0 Common Final Review Questions In problems 1 6, perform the indicated operations and simplif if necessar. 1. ( 8)(4) ( )(9) 4 7 4 6( ). 18 6 8. ( i) ( 1 4 i ) 4. (8 i ). ( 9 i)( 7 i) 6. ( i)( i )
More information(2) Find the domain of f (x) = 2x3 5 x 2 + x 6
CHAPTER FUNCTIONS AND MODELS () Determine whether the curve is the graph of a function of. If it is state the domain and the range of the function. 5 8 Determine whether the curve is the graph of a function
More informationA11.1 Areas under curves
Applications 11.1 Areas under curves A11.1 Areas under curves Before ou start You should be able to: calculate the value of given the value of in algebraic equations of curves calculate the area of a trapezium.
More informationReview Topics for MATH 1400 Elements of Calculus Table of Contents
Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical
More informationSections 4.1 & 4.2 Exponential Growth and Exponential Decay
8 Sections 4. & 4.2 Eponential Growth and Eponential Deca What You Will Learn:. How to graph eponential growth functions. 2. How to graph eponential deca functions. Eponential Growth This is demonstrated
More informationDerivatives of Multivariable Functions
Chapter 0 Derivatives of Multivariable Functions 0. Limits Motivating Questions In this section, we strive to understand the ideas generated b the following important questions: What do we mean b the limit
More informationPrerequisites for Calculus
CHAPTER Prerequisites for Calculus. Lines. Functions and Graphs.3 Eponential Functions.4 Parametric Equations.5 Functions and Logarithms Eponential functions are used to model situations in which growth
More informationCHAPTER 3 Exponential and Logarithmic Functions
CHAPTER Eponential and Logarithmic Functions Section. Eponential Functions and Their Graphs......... Section. Logarithmic Functions and Their Graphs......... Section. Properties of Logarithms..................
More informationSpeed (km/h) How can you determine the inverse of a function?
.7 Inverse of a Function Engineers have been able to determine the relationship between the speed of a car and its stopping distance. A tpical function describing this relationship is D.v, where D is the
More information5A Exponential functions
Chapter 5 5 Eponential and logarithmic functions bjectives To graph eponential and logarithmic functions and transformations of these functions. To introduce Euler s number e. To revise the inde and logarithm
More informationFair Game Review. Chapter 8. Graph the linear equation. Big Ideas Math Algebra Record and Practice Journal
Name Date Chapter Graph the linear equation. Fair Game Review. =. = +. =. =. = +. = + Copright Big Ideas Learning, LLC Big Ideas Math Algebra Name Date Chapter Fair Game Review (continued) Evaluate the
More information2.1 The Rectangular Coordinate System
. The Rectangular Coordinate Sstem In this section ou will learn to: plot points in a rectangular coordinate sstem understand basic functions of the graphing calculator graph equations b generating a table
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions. Eponential Functions and Their Graphs. Logarithmic Functions and Their Graphs. Properties of Logarithms. Eponential and Logarithmic Equations.5 Eponential and Logarithmic
More informationExact Differential Equations. The general solution of the equation is f x, y C. If f has continuous second partials, then M y 2 f
APPENDIX C Additional Topics in Differential Equations APPENDIX C. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Chapter 6, ou studied applications
More informationThe American School of Marrakesh. Algebra 2 Algebra 2 Summer Preparation Packet
The American School of Marrakesh Algebra Algebra Summer Preparation Packet Summer 016 Algebra Summer Preparation Packet This summer packet contains eciting math problems designed to ensure our readiness
More information5Higher-degree ONLINE PAGE PROOFS. polynomials
5Higher-degree polnomials 5. Kick off with CAS 5.2 Quartic polnomials 5.3 Families of polnomials 5.4 Numerical approimations to roots of polnomial equations 5.5 Review 5. Kick off with CAS Quartic transformations
More informationa b a b ab b b b Math 154B Elementary Algebra Spring 2012
Math 154B Elementar Algera Spring 01 Stud Guide for Eam 4 Eam 4 is scheduled for Thursda, Ma rd. You ma use a " 5" note card (oth sides) and a scientific calculator. You are epected to know (or have written
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.
More informationMathematics Background
UNIT OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND UNIT INTRODUCTION Patterns of Change and Relationships The introduction to this Unit points out to students that throughout their study of Connected
More informationEssential Question How can you use a quadratic function to model a real-life situation?
3. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..E A..A A..B A..C Modeling with Quadratic Functions Essential Question How can ou use a quadratic function to model a real-life situation? Work with a partner.
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson, you Learn the terminology associated with polynomials Use the finite differences method to determine the degree of a polynomial
More informationHigher. Functions and Graphs. Functions and Graphs 15
Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 5 Set Theor 5 Functions 6 Inverse Functions 9 4 Eponential Functions 0 5 Introduction to Logarithms 0 6 Radians 7 Eact Values
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions Eponential functions are those with variable powers, e.g. = a. Their graphs take two forms: (0, 1) (0, 1) When a > 1, the graph: is alwas increasing is alwas positive
More informationLESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II
LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will
More informationSchool of Business. Blank Page
Equations 5 The aim of this unit is to equip the learners with the concept of equations. The principal foci of this unit are degree of an equation, inequalities, quadratic equations, simultaneous linear
More information12Variation UNCORRECTED PAGE PROOFS
Variation. Kick off with CAS. Direct, inverse and joint variation. Data transformations. Data modelling. Review U N C O R R EC TE D PA G E PR O O FS. Kick off with CAS Please refer to the Resources ta
More informationragsdale (zdr82) HW7 ditmire (58335) 1 The magnetic force is
ragsdale (zdr8) HW7 ditmire (585) This print-out should have 8 questions. Multiple-choice questions ma continue on the net column or page find all choices efore answering. 00 0.0 points A wire carring
More informationChapters 8 & 9 Review for Final
Math 203 - Intermediate Algebra Professor Valdez Chapters 8 & 9 Review for Final SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the formula for
More informationLESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II
LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS
More informationItems with a symbol next to the item number indicate that a student should be prepared to complete items like these with or without a calculator.
HNRS ALGEBRA B Semester Eam Review The semester B eamination for Honors Algebra will consist of two parts. Part is selected response on which a calculator will NT be allowed. Part is short answer on which
More information) approaches e
COMMON CORE Learning Standards HSF-IF.C.7e HSF-LE.B.5. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. The Natural
More informationFinal Exam Review. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Final Eam Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function.
More information14.1 Systems of Linear Equations in Two Variables
86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination
More informationPrecalculus Honors - AP Calculus A Information and Summer Assignment
Precalculus Honors - AP Calculus A Information and Summer Assignment General Information: Competenc in Algebra and Trigonometr is absolutel essential. The calculator will not alwas be available for ou
More informationConic Sections CHAPTER OUTLINE. The Circle Ellipses and Hyperbolas Second-Degree Inequalities and Nonlinear Systems FIGURE 1
088_0_p676-7 /7/0 :5 PM Page 676 (FPG International / Telegraph Colour Librar) Conic Sections CHAPTER OUTLINE. The Circle. Ellipses and Hperbolas.3 Second-Degree Inequalities and Nonlinear Sstems O ne
More informationAnswers to All Exercises
Answers to All Eercises CHAPTER 5 CHAPTER 5 CHAPTER 5 CHAPTER REFRESHING YOUR SKILLS FOR CHAPTER 5 1a. between 3 and 4 (about 3.3) 1b. between 6 and 7 (about 6.9) 1c. between 7 and 8 (about 7.4) 1d. between
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationLESSON 12.2 LOGS AND THEIR PROPERTIES
LESSON. LOGS AND THEIR PROPERTIES LESSON. LOGS AND THEIR PROPERTIES 5 OVERVIEW Here's what ou'll learn in this lesson: The Logarithm Function a. Converting from eponents to logarithms and from logarithms
More informationREVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES
Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.
More informationSystems of Equations and Inequalities
Sstems of Equations and Inequalities 7 7. Linear and Nonlinear Sstems of Equations 7. Two-Variable Linear Sstems 7.3 Multivariable Linear Sstems 7. Partial Fractions 7.5 Sstems of Inequalities 7.6 Linear
More information4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
More informationLESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II
LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS
More information11.1 Inverses of Simple Quadratic and Cubic Functions
Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A,
More informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More information5 Linear Graphs and Equations
Linear Graphs and Equations. Coordinates Firstl, we recap the concept of (, ) coordinates, illustrated in the following eamples. Eample On a set of coordinate aes, plot the points A (, ), B (0, ), C (,
More informationInverse of a Function
. Inverse o a Function Essential Question How can ou sketch the graph o the inverse o a unction? Graphing Functions and Their Inverses CONSTRUCTING VIABLE ARGUMENTS To be proicient in math, ou need to
More informationPRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line.
MAC 1105 PRACTICE FINAL EXAM College Algebra *Note: this eam is provided as practice onl. It was based on a book previousl used for this course. You should not onl stud these problems in preparing for
More informationMath RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus
Math 0-03-RE - Calculus I Functions Page of 0 Definition of a function f() : Topics of Functions used in Calculus A function = f() is a relation between variables and such that for ever value onl one value.
More information(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
More information