17 Exponential Functions

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1 Eponential Functions Concepts: Eponential Functions Graphing Eponential Functions Eponential Growth and Eponential Deca The Irrational Number e and Continuousl Compounded Interest (Section. &.A). Sketch a complete graph of each of the following: (a) (b) (c) e. (Question, Section.) Match the functions to the graphs. Assume a >. (a) graph A = (b) graph B = (c) graph C = (d) graph D =. Describe the transformations necessar for producing each graph from the graph of f() = e. (a) g() = e (b) h() = e (c) k() = e 4. If $0,000 is invested at an interest rate of % per ear, find the amounts in the account after ears if interest is compounded (a) quarterl (b) monthl (c) dail

2 . If $0,000 is invested at an interest rate of 4% per ear, compounded quarterl, find the value of the investment after the given number of ears. (a) ears (b) 0 ears (c) ears (d) 0 ears 6. Joni invests $000 at an interest rate of % per ear compounded continuousl. How much time will it take for the value of the investment to quadruple.. Between 90 and 860, the population of the United States (in millions) in the ear was given b =.9(.099 ), where is the number of ears since 90. Find the population in the ear 88. (The question similar to this on Webassign epects our answer to be epressed as an actual number, not in units of,000,000.) 8. According to the Kell Blue Book, the factor invoice price of a 00 Buick LaCrosse 4-door CX Sedan is $,94. and the MSRP is $6, (The factor invoice price is the price the dealer pas to the factor. The MSRP is the Manufacturer s Suggested Retail Price.) A certain dealership is charging the MSRP for the LaCrosse, but the need to clear their inventor to make room for the 0 models. The decide to reduce the price of the car b % ever da after October, 00. (On October, the reduce the price b %; on October, the reduce the reduced price b %; etc.) (a) What will the price of the car be on October, 00? (b) Find a function that models the price p of the car t das after October, 00. (c) Will the car ever be free? (d) On what date does the dealer start to lose mone? 9. The graph of g is shown below. Find a formula for g()

3 0. Label each of the following graphs with at least one of the following categories: Linear, Quadratic, Polnomial, Rational, Eponential, or None of the Categories Studied So Far. Some graphs ma have more than one label.

4 8 Logarithmic Functions Concepts: Logarithms as Functions Logarithms as Eponent Pickers Inverse Relationship between Logarithmic and Eponential Functions. Properties of Logarithms (Sections. &.4). Find the eact value of the following logarithms. Do NOT use our calculator. (a) log () (b) log( ( ) 00) (c) 0 log() (d) e ln() (e) ln e. Using our knowledge of eponents, estimate between which two integer values the following epressions will be. Use our calculator to find approimate values for each. Was our estimation accurate? (a) log(008) (b) ln() (c) ln() (d) log(). Convert each eponential statement to an equivalent logarithmic statement. (a) 8 = 64 (b) = 6 (c) 4 = (d) = ( ) 4. Convert each logarithmic statement to an equivalent eponential statement. (a) log () = (b) ln() = (c) log(9) = (d) log ( 64 ) =. (Questions 49 & 0, Section.) Do the graphs of following functions appear to be the same? How do the differ? (a) f() = log( ) and g() = log() (b) h() = log( ) and k() = log() 6. Find all real solutions or state that there are none. Your answers should be eact. (a) ln() = (b) 0 + = 6 (c) 9e 8 = (d) log 8 ( ) log 8 ( + ) = 0

5 . Given the following functions, find f (). (a) f() = ln( + ) (b) f() = + 8. Given the following functions, find the domain. (a) f() = ln( ) (b) g() = ln( + 4) (c) h() = ln( ) 9. What transformations are needed to find the graph of each function from the graph of f() = ln()? (a) g() = ln() (b) h() = ln() (c) k() = ln( ) (d) l() = ln(+) 0. (Question # 8, Section.) f() = A ln() + B, where A and B are constants. If f(e) = and f(e ) = 8, what are A and B?. (Question # 8, Section.) Students in a precalculus class were given a final eam. Each month thereafter, the took an equivalent eam. The class average on the eam taken after t months is given b: F (t) = 8 8 ln(t + ). (a) What was the class average after si months? (b) After a ear? (c) When did the class average drop below?. Write each epression as a single logarithm. If it is not possible, eplain wh. (a) (ln()) 4 (ln( ) + ln()) (b) log ( ) + (ln( ) ln()) (c) (ln(e e)). Write each epression in terms of log(), log(), and log(z). If it is not possible, eplain wh. ( ) ( ) + (a) log (b) log (c) log ( z ) z z ( u ) 4. Prove the Quotient Rule for Logarithms, log a = log v a (u) log a (v).

6 9 Logarithmic Properties & Equations Concepts: Using the Change of Base Formula to Find Approimate Values of Logarithms Solving Eponential and Logarithmic Equations Algebraicall Strategies: Same base eponential epressions that are equal must have equal eponents. Same base logarithmic epressions that are equal must have equal arguments. Isolate eponential epression, rewrite equation in logarithmic form. Isolate logarithmic epression, rewrite equation in eponential form. Solving Eponential and Logarithmic Models/Applications (Sections.4 -.). Use our calculator to find approimate values for the following. (a) ln() (b) log() (c) log (6) (d) log (). Given the magnitude of an earthquake on the Richter scale is given b R(i) = log( i i 0 ), where i is the amplitude of the ground motion of the earthquake and i 0 is the amplitude of the ground motion of the zero earthquake, (a) Find the magnitude on the Richter scale of an earthquake that is 0000 times stronger than the zero quake. (b) Find the magnitude on the Richter scale of an earthquake that is times stronger than the zero quake.. The half-life of a certain radioactive substance is,6 ears. (a) Find the deca rate constant r. (b) How much substance will be left in 00 ears if there is currentl 00 grams of the substance? 4. The antibiotic clarithromcin is eliminated from the bod according to the formula A(t) = 00e 0.86t, where A is the amount remaining in the bod (in milligrams) t hours after the drug reaches peak concentration. (a) How much time will pass before the amount of drug in the bod is reduced to 00 milligrams? (b) Find the inverse of A(t) and eplain what the inverse function models.

7 . You are given models for the population (in millions) of different countries t ears after 00. For each part, determine the ear in which the models predict the populations will be equal. (Source: World Health Organization s 006 World Health Statistics) (a) Rwanda: R(t) = 9.04(.0) t and Hungar: H(t) = 0.(0.98) t. (b) Cambodia: C(t) = 4.0(.0) t and Kazakhstan: K(t) = 4.8(0.9) t. 6. Find the solution(s) of the following eponential equations. Your answers should be eact. (a) 0 = 0 9 (b) + = 0 (c) + e = (d) 4 6 = 0 (e) 9e 8 =. Find the solution(s) of the following logarithmic equations. Your answers should be eact. (a) log 4 ( + ) + log 4 = log 4 + log 4 ( ) (b) log ( + ) log ( ) = (c) log (log ) = 4 (d) log( + ) = log + log (e) log 8 ( ) + log 8 ( + ) = 8. Suppose ou re driving our car on a cold winter da (0 F outside) and the engine overheats (at about 0 F). When ou park, the engine begins to cool down. The temperature U of the engine t minutes after ou park satisfies the equation ( ) U 0 ln = 0.t. 00 (a) Solve the equation for U. (b) Use part (a) to find the temperature of the engine after 0 min (t = 0). 9. Find how long it takes for a deposit to double in value if the annual interest rate is.% and the interest is compounded continuousl. 0. Joni invests $000 at an interest rate of % per ear compounded continuousl. How much time will it take for the value of the investment to quadruple?. Joni invests $000 at an interest rate of % per ear compounded monthl. How much time will it take for the value of the investment to quadruple?

8 0 Angles, Their Measurement, and Basic Trig Functions Concepts: Angles Initial Side, Terminal Side, Standard Position & Coterminal Angles Measuring Angles Radian Measure vs. Degree Measure Radian Measure as a Distance on the Unit Circle Converting between Radian Measure and Degree Measure Finding the Quadrant Associated with the Terminal Side of an Angle Identifing the Point on the Unit Circle that Corresponds to an Angle in Standard Position The Trigonometric Functions The Definitions of sin, cos, and tan Based on the Unit Circle Evaluating the Basic Trigonometric Functions at Special Angles The Sign of a Trigonometric Function The π π π or the Triangle The π π π or the Triangle (Sections 6. & 6.). Find the radian measure of each of the following: (a) 40 angle (b) 0 angle. Show which of the following points must lie on the unit circle. (a) (0, ) (b) (, ) (e) (, ) (f) (, ) (c) (, 4) (g) (, ) (d) ( 4, ) (h) (, ) ( ). Suppose than an angle of measure θ radians intersects the unit circle at the point,. (a) What is one possibilit for θ? (b) How do ou find all the other possibilities?

9 4. Suppose than an angle of measure θ radians intersects the unit circle at the point (a) What is one possibilit for θ? (b) How do ou find all the other possibilities? ( ),.. Suppose that an angle of measure θ radians is placed in standard position. Find the location of the terminal side of the angle. Possibilities: (A) Quadrant I, (B) Quadrant II, (C) Quadrant III, (D) Quadrant VI, (E) the positive -ais, (F) the negative -ais, (G) the positive -ais, or (H) the negative -ais. (a) θ = 4π (b) θ = 4π (c) θ = 00π (d) θ = 00π (e) θ = π (f) θ = π (g) θ = 0π (h) θ = 0π 6. Find the terminal point on the unit circle determined b the given value of θ. (a) θ = 4π (b) θ = π (c) θ = π 6 (d) θ = π 6 (e) θ = π 4 (f) θ = π (g) θ = 4π (h) θ = π 6. Evaluate the basic trigonometric functions at each of the following angles. (a) θ = π (b) θ = 9π 4 (c) θ = 4π (d) θ = π 6 8. (a) The terminal side of an angle, θ, in standard position contains the point (, 9). Evaluate the basic trigonometric functions at θ. (b) The terminal side of an angle, θ, in standard position contains the point (, 4). Evaluate the basic trigonometric functions at θ. 9. Suppose θ is in the fourth quadrant and cos(θ) =. Evaluate the remaining basic trigonometric functions on θ. 0. Suppose θ is in the first quadrant and sin(θ) = 6. Find each of the following (give eact answers): (a) sin(8π + θ) (b) tan( θ) (c) cos(4π θ)

10 Introduction to Trigonometric Graphs Worksheet Concepts: Understanding Trigonometric Notation The Graph of the sin, cos, and tan Functions Appling Graph Transformations to the Graphs of the sin, cos, and tan Functions Using Graphical Evidence to Make Conjectures about Identities (Sections 6. & 6.4). (Question 4, Section 6.) When a plane flies faster than the speed of sound, the sound waves it generates trail the plane in a cone shape, as shown in the figure in the Section 6. eercises of our tetbook. When the bottom part of the cone hits the ground ou hear a sonic boom. The equation that describes this situation is ( ) t sin = w p where t is the radian measure of the angle of the cone, w is the speed of the the sound wave, p is the speed of the plane, and p > w. (a) Find the speed of the sound wave when the plane flies at 00 mph and t =.8. (b) Find the speed of the plane if the sound wave travels at 00 mph and t =... (Question 8, Section 6.) Given that the terminal side of an angle of t radians in standard position lies in Quadrant III on the line = 0, find sin(t), cos(t), tan(t). HINT: Find a point on the terminal side of the angle.. Find all the solutions to the following equations: (a) cos(t) = (b) tan(t) = 0 (c) sin(t) = 4. For each of the following equations, list the transformations that ou need to appl to the graph of = sin(), = cos(), or = tan() to sketch the graph of the equation. Sketch the graph. Be sure that the graph is well-labeled. (a) = 4 sin () ( ) (b) = cos π ( (c) = sin π ) ( (d) = tan + π ) 4 (e) = u cos(v + w). (Assume that u, v, and w are positive.)

11 . Use graphical evidence to determine which of the following MIGHT be trigonometric identities and which definitel cannot be trigonometric identities. (a) cos() = cos () sin () (b) sin() = sin () cos () (c) sin( + ) = sin() + sin() (d) sin() cos() = (sin( + ) + sin( )) (e) tan(θ) = tan() tan(θ) (f) sin(t ) t = sin(t) 6. The graph of a trigonometric function is shown below. Find a rule for the function. π π π 4π. The graph of a trigonometric function is shown below. Find a rule for the function. π π π 4π

12 More Trigonometric Graphs Worksheet Concepts: Period, Amplitude, and Phase Shift The csc, sec, and cot Functions The Graphs of the csc, sec, and cot Functions (Sections 6. and 6.6). For each graph, (i) find the period if it is defined and (ii) find the amplitude if it is defined. (a) (b) For each of the following equations, what is the period, amplitude and phase shift of the graph? (Assume that u, v, and w are positive.) ( ) ( (a) = cos (c) = tan + π ) ( π (b) = sin π ) 4 (d) = u cos(v + w).

13 . (Question 48 from Section 6. of our tetbook) The number of hours of dalight in Winnipeg, Manitoba, can be approimated b d(t) = 4. sin(.0t.) +, where t is measured in das, with t = being Januar. (a) On what da is there the most dalight? The least? How much dalight is there on these das? (b) On which das are there hours or more of dalight? What do ou think the period of this function is? Wh? 4. Evaluate the csc, sec, and cot functions at each of the following angles. (a) θ = π (b) θ = 9π 4 (c) θ = 4π (d) θ = π 6. (a) The terminal side of an angle, θ, in standard position contains the point (, 9). Evaluate the csc, sec, and cot functions at θ. (b) The terminal side of an angle, θ, in standard position contains the point (, 4). Evaluate the csc, sec, and cot functions at θ. 6. Suppose θ is in Quad IV and cos(θ) =. Find csc(θ), sec(θ), and cot(θ).. Sketch the graphs of the following equations. Be sure that the graph is well-labeled. (a) = csc () + ( ) (b) = sec + ( (c) = cot π ) 4 8. Use algebra and identities to simplif the epression. Assume all denominators are nonzero. (a) sin(t) tan(t) 9. Solve each of the following equations. (a) cos()=0 (b) sin (t) sin(t) = 0 (c) cos (t) cos(t) = (b) cos(t) sin(t)tan(t)

14 Trigonometric Identities Worksheet Concepts: Trigonometric Identities Reciprocal Identities Pthagorean Identities Periodicit Identities Negative Angle Identitites (Sections 6.6 &.). Find the period, phase shift, and sketch the graph of = csc( π ).. Find all solutions to the equation tan(θ) = cot(θ) for 0 < θ < π. Use the Pthagorean identit for each of the following problems. (a) Find sin(t) if cos(t) = 0 and π < t < π. (b) Find cos(t) if sin(t) = and π < t < π. (c) Find cot(t) if sin(t) = and π < t < π. 4. Determine each of the following: (a) Find cos if sin = and tan > 0. (b) Find tan if cos = 4 and sin < 0. (c) Find tan if sec = 0 and sin > 0.

15 . Use the periodicit and negative angle identities to solve each of the following problems. (a) Find sin( t) if sin(t) =. (b) Find sin(t π) if sin(t) =. (c) Find cos( t + 4π) if cos(t) = 6. (d) Find cot(π t) if tan(t) =. 6. Use basic identities to simplif the epression. (a) cot θ sec θ sin θ (b) cos sin + csc sin (c) + sec cos cot (d) sin + tan + cos sin (e) sin csc cos (f) sin + sin cos (g) Prove the identit sin (tan cos cot cos ) = cos. State whether or not the equation might be an identit. If it appears to be, prove it. (a) cot = csc sec (b) sin( ) cos( ) = tan (c) + sec = tan (d) sin (cot + ) = cos (tan + ) (e) + sin sec + tan = sin sec tan (f) ( cos ) csc = sin (g) sec csc + sin cos = tan (h) csc = cos csc sin cos tan (i) = tan sin + cos

16 4 MA 0 Eam Practice Worksheet Sections. -., , ,. Do not rel solel on this practice eam! Make sure to stud homework problems, other work sheets, lecture notes, and the book!!!. If $,000 is deposited in a bank account which has a earl interest rate of r =.% compounded continuousl, find how much is in the account after ears and months. (a) $. (b) $. (c) $9.88 (d) $49. (e) $ Find the domain of f() = ln( + + ). (a) (, ) (, ) (b) (, ) (c) [0, ) (d) (, ) (e) (0, ). Rewrite the following epression as a single logarithm. log a log a + 6 log a w log a z. (a) log a z w /6 / (b) log a w /6 / z (c) log a / w /6 z (d) log a w /6 / z (e) log a ( + 6 w z) 4. Find the radian measure of a standard position angle between 0 and π that is coterminal with π 6. (a) π 6 (b) π 6 (c) π 6 (d) π 6 (e) π 6

17 . Find the point on the unit circle corresponding to the angle π radians. 6 ( (a) ), ( (b), ) (c) (0, ) (d) ( (e) (, 0), ) 6. Determine the number of solutions between 0 and 6π of the following equation. cos(t) = (a) (b) (c) (d) 4 (e) 6. Find the value of cot(t) given that sin(t) = 8 and sec(t) < 0. (a) (b) 8 (c) 8 (d) (e) 8 8. Solve the equation for. log () = log ( 6) + 9. Consider the following graph of the form f(t) = A sin(bt + c). (a) Find the amplitude, period, and phase shift. (b) Find A, b, and c and state the rule of the function. 0. Prove the following identit: sin(t) cos(t) + cos(t) sin(t) = csc(t)

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