f(x) = d(x) q(x) + r(x).

Transcription

1 Section 5.4: Dividing Polynomials 1. The division algorithm states, given a polynomial dividend, f(x), and non-zero polynomial divisor, d(x), where the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that f(x) = d(x) q(x) + r(x). where q(x) is quotient and r(x) is the remainder. (The remainder is either equal to zero or has degree strictly less than d(x).) If r(x) 0 then d(x) divides evenly into f(x). This means that both d(x) and q(x) are factors of f(x). 2. Divide 2x 3 3x 2 + 4x + 5 by x Use long division to simplify: 5x2 + 3x 2. x + 1 1

2 4. Use long division to find the dividend, divisor, quotient and remainder of 6x x 2 31x + 15 divided by 3x Synthetic division is a shortcut that can be used when the divisor is a binomial in the form x k where k is a real number. In synthetic division, only the coefficients are used in the division process. 6. Use synthetic division to divide 5x 2 3x 36 by x 3. 2

3 7. Use synthetic division to divide 4x x 2 6x 20 by x + 2 and use this information to graph f(x) = 4x x 2 6x The volume of a rectangular solid is given by the polynomial 3x 4 3x 3 33x x. The length of the solid is given by 3x and the width is given by x 2. Find the height, t of the solid. 3

4 Section 5.5: Zeros of Polynomial Functions 1. If a polynomial f(x) is divided by x k, then the remainder is the value f(k). 2. Use the remainder theorem to evaluate f(x) = 6x 4 x 3 15x 2 + 2x 7 at x = According to the Factor Theorem, k is a zero of f(x) if and only if x k is a factor of f(x). 4. Show that x + 2 is a factor of x 3 6x 2 x Find the remaining factors. Use the factors to determine the zeros of the polynomial. 5. The rational zero theorem states that, if the polynomial f(x) = a n x n + a n 1 x n a 1 x + a 0 has integer coefficients, then every rational zero of f(x) has the form p where p is a factor of the q constant term a 0 and q is a factor of the leading coefficient a n. When the leading coefficient is 1, the possible zeros are the factors of the constant term. 6. List all possible rational zeros of f(x) = 2x 4 5x 3 + x

5 7. Use the rational zero theorem to find the rational zeros of f(x) = 2x 3 + x 2 4x Find the zeros of 4x 3 3x 1 and graph it! 9. The Fundamental Theorem of Algebra states that, if f(x) is a polynomial of degree n > 0, then f(x) has at least one complex zero. We can use this theorem to argue that, if f(x) is a polynomial of degree n > 0, and a is a non-zero real number, then f(x) has exactly n linear factors f(x) = a(x c 1 )(x c 2 ) (x c n ) where c 1, c 2,..., c n are complex numbers. Therefore, f(x) has n roots if we allow for multiplicities. 2

6 10. Find the zeros of f(x) = 3x 3 + 9x 2 + x According to the Linear Factorization Theorem, a polynomial function will have the same number of factors as its degree, and each factor will be in the form (x c) where c is a complex number. If the polynomial function f has real coefficients and a complex zero in the form a + bi, then the complex conjugate a bi is also a zero. 12. State a fourth degree polynomial with real coefficients that has zeros of 3, 2, i such that f( 2) =

7 Section 5.6: Rational Functions 1. Draw the graphs of 1 x and 1, taking note of different properties. x2 2. Vertical Asymptote: a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a. We write As x a, f(x), or as x a, f(x). (Note the vertical asymptotes in 1 x and 1 x 2.) Horizontal Asymptote: a horizontal line y = b where the graph approaches the line as the inputs increase or decrease without bound we write: as x or x, f(x) b. 3. Describe the end behavior and local behavior of the following graph, and write a possible function for the graph. 1

8 4. A rational function is a function that can be written as the quotient of two polynomial functions, p(x) and q(x) where q(x) 0. f(x) = p(x) q(x) = a px p + a p 1 x p a 1 x + a 0 b q x q + b q 1 x q b 1 x + b 0. Note, the domain of a rational function includes all real numbers except those that cause the denominator to be zero. 5. A large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time the sugar is poured into the tank at a rate of 1 pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after 12 minutes. Is that a greater concentration than at the beginning? 6. Find the domain of f(x) = x + 3 x (See page 420 for steps, if needed) Find the vertical asymptotes of the graph of k(x) = 5 + 2x2 2 x x 2 8. Find the removable discontinuity (hole) of f(x) = x 2 1x 2 2x 3 2

9 9. Find the vertical asymptotes, removable discontinuities and graph f(x) = x 2 x Find the horizontal/slant asymptotes of the rational functions. (a) f(x) = 4x + 2 x 2 + 4x 5 (b) f(x) = x2 4x + 1 x + 2 (c) f(x) = 3x2 + 2 x 2 + 4x 5 3

10 11. Consider the function: f(x) = (x 2)(x + 3) (x 1)(x + 2)(x 5) (a) Are there any holes, removable discontinuities? If so what are they? (b) Find all asymptotes of the function. (c) Find all intercepts of the function. 12. Write the equation for the rational function that has vertical asymptotes at x = ±5, x-intercepts at (2, 0) and ( 1, 0), and y-intercept at (0, 4). 13. Write the equation for the rational function that has vertical asymptote at x = 1, double zero at x = 2, y-intercept at (0, 2). 4

11 Section 5.8: Modeling Using Variation 1. A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Write a function of her earnings, based upon her sales. 2. Direct Variation: If x and y are related by an equation of the form y = kx n then we say that y varies directly with the nth power of x (or the relationship is direct variation). k is called the constant of variation. 3. y varies directly with the cube of x. If y = 25 when x = 2, find y when x is Water temperature in an ocean varies inversely to the water s depth. Between the depths of 250 feet and 500 feet, the formula T = 14,000 gives us the temperature in degrees Fahrenheit at a depth d in feet below Earth s surface. At a depth of 2000 feet what is the water temperature? What about 500 feet? 5. Inverse variation: y varies inversely (or inversely proportional) with the nth power of x if y = k x n. 1

12 6. A tourist plans to drive 100 miles. Find the formula for the time the trip will take as a function of the speed the tourist drives. 7. A quantity y varies inversely with the cube of x. If y = 25 when x = 2, find y when x is Joint Variation: If x varies directly with both y and z we have x = kyz. If x varies directly with y and inversely with z we have x = ky. (We only use one constant in joint variation.) z 9. A quantity x varies directly with the square of y and inversely with the cube root of z. If z = 6 when y = 2 and z = 8, find x when y = 1 and z is Find the variation constant and an equation of variation in which y varies directly as x and y = 32 when x = 2. 2

13 11. The number of centimeters of water W produced from melting snow varies directly as S, the number of centimeters of snow. Meteorologists have found that under certain conditions 150 cm of snow will melt to 16.8 cm of water. To how many centimeters of water will 200 cm of snow melt under the same conditions? 12. Find the variation constant and an equation of variation in which y varies inversely as x and y = 16 when x = The time t required to fill a swimming pool varies inversely as the rate of flow r of water into the pool. A tank truck can fill a pool in 90 minutes at a rate of 1500L/min. How long would it take to fill the pool at a rate of 1800L/min? 3

14 14. The volume of wood V in a tree varies jointly as the height h and the square of the girth g. (Girth is distance around.) If the volume of a redwood tree is 216m 3 when the height is 30m and the girth is 1.5m, what is the height of a tree whose volume is 960m 3 and whose girth is 2m? 15. The volume V of a given mass of a gas varies directly as the temperature T and inversely as the pressure P. If V = 231 cm 3 when T = 42 and P = 20kg/cm 2, what is the volume when T = 30 and P = 15kg/cm 2? 4

15 Section 6.1: Exponential Functions 1. India is the second most populous country in the world with a population of about 1.25 billion people in The population is growing at a rate of about 1.2% each year. What will the population be in 2014? 2015? A city, Maple Valley s population is growing by 124 people per year. If there were 25,125 people in 2014, what is the population in 2015? 2016? 2. Exponential Function terms : Percent change refers to a change based on a percent of the original amount. Exponential growth refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time. Exponential decay refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time. Notice the differences between an exponential function and a linear function below by completing the table: x f(x) = 2 x g(x) = 2x

16 3. Exponential Function For any real number x, an exponential function is a function with the form f(x) = ab x (draw two examples, growth and decay, beside the information) where a is called the initial value and is a non-zero real number b is any positive real number such that b 1 Domain of f is all real numbers Range of f If a > 0 then all positive real numbers If a < 0 then all negative real numbers y-intercept is (0, a) and horizontal asymptote is y = 0. Exponential Growth/Decay: Exponential Growth: when a > 0 and b > 1. a is the initial, or starting value and b is the growth factor or growth multiplier per unit x. Exponential Decay: (talking about applied problems we usually have the fact that a > 0) ALWAYS 0 < b < 1. a is the initial, or starting value and b is the growth factor or growth multiplier per unit x. There is a special exponential function called a continuous growth/decay function a is the initial value t is the elapsed time f(x) = ae rt r is the continuous growth rate per unit time r > 0 then continuous growth r < 0 then continuous decay 4. Let f(x) = 5(3) x+1 evaluate f(2), f( 1) and f(0). 2

17 5. India is the second most populous country in the world with a population of about 1.25 billion people in The population is growing at a rate of about 1.2% each year. Write the population, P (t) in billions of people where t is years since What is the population in 2017 according to this model? 6. When creating exponential functions, unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places for the remainder of this section! In 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had grown to 180 deer. If the population was growing exponentially, write an algebraic fuction E(t) representing the population E of deer in years since If the population was growing linearly, write an algebraic function L(t) representing the population L of deer in years since

18 7. Find an exponential funciton that passes through the points ( 2, 6) and (2, 1). 8. Find the equation for the exponential function graphed below: 9. Compound Interest A = P ( 1 + r ) nt n where A-account value, future value t-years P -starting value, principal or present value r-annual percentage rate (APR) expressed as a decimal n-number of compounding periods in one year. Continuous Compound Interest (with the same definitions as above!) A = P e rt 4

19 10. If we invest $3000 in an account with 3% compounded quarterly, how much will the account be worth in 10 years? 11. A 529 Plan is a college-savings plan that allows relatives to invest money wo pay for a child s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in teh account now? 12. A person invested $1000 in an account earning a nominal 10% per year compounded continuously. How much was in the account at the end of one year? 13. Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days? 5

20 Section 6.2: Graphs of Exponential Functions Sketch a graph of f(x) = 0.25 x. State the domain, range and asymptote. 3. 1

21 4. Graph f(x) = 2 x+1 3 and state the domain, range and asymptote Sketch a graph of f(x) = 4 ( 1 2) x. State the domain, range and asymptote. 7. 2

22 8. Write the function and graph g(x), which reflects f(x) = ( 1 4) x about the x-axis. State domain, range and asymptote. 9. Summary of transformations: 10. Write the equation for the function described. f(x) = e x is vertically stretched by a factor of 2, reflected across the y-axis and shifted up 4 units. 3

23 Section 6.3: Logarithmic Functions Section 6.4: Graphs of Logarithmic Functions 1. A logarithm base b of a positive number x satisfies the following definition: log b (x) = y is equivalent to b y = x, where x, b > 0, b 1. where if b = 10 this is the common logarithm and is written log(x). if b = e this is the natural logarithm and is written ln(x). range of log is (, ) domain of log is (0, ). [Note this says you cannot take the log of a negative number!] A few properties (will repeat in section 6.5): log a 1 = 0 log a a = 1 log 10 x = log x log e x = ln x log b M = log a M log a b for all a, b and M > Convert the following to a logarithm: (a) 16 = 2 x (b) 10 3 = (c) e t = 70 1

24 3. Convert to an exponential (a) log 2 32 = 5 (b) log a Q = 8 (c) log t M = x 4. Use calculator to solve (round to two decimals as needed): (a) log 645, 778 (d) ln 645, 778 (b) log (e) ln e (c) log( 3) (f) ln 1 (g) Find log 5 8 using common logarithm. 5. The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation 10 x = 500 represents this situation, where x is the difference in magnitudes on teh Richter Scale. To the nearest thousandth, what was the difference in magnitudes? 2

25 6. Earthquake Intensity, I, measured on a Richter Scale of magnitude R is R = log I I 0 where I 0 is the minimum intensity to be measured. On March 11, 2011 there was an earthquake of intensity of I 0 off the coast of Japan. What is the magnitude of this earthquake? 7. On November 1, 2017 at 1:19PM about 8km from Hennessey, OK there was an earthquake of magnitude 3.5 what was the intensity of this earthquake? 8. Average walking speed, w in feet per second of a person living in a city of population P, in thousands, is given by w(p ) = 0.37 ln P (a) Population in Savannah, Georgia is 136, 286 find average walking speed. (b) Population in Philadelphia is 1, 526, 006 find the average walking speed. (c) The average walking speed of a city is about 2.0 ft/sec, what is the approximate population of the city. 3

26 6.4: Graphs: 9. Graph e x and it s inverse ln(x) on the same axis. List domain/range of both functions. 10. Without graphing f(x) = log 2 (x + 3) what is the domain of this function? 11. 4

27 12. Graph the following: (a) ln(x + 3) (b) ln x (c) ln(x 1) 5

28 Section 6.5: Logarithmic Properties Properties of Logarithms: Assume A and B are positive real numbers, a, b > 0, a, b 1 and p is any real number. Properties of Logarithms: log a (A B) = log a A + log a B log a (A/B) = log a A log a B log a A p = p log a A log b M = log a M log a b log a a = 1 log a 1 = 0 a log a x = x log a a x = x 1. Expand, using the properties of logarithms: (a) log xy z (b) log 2x 1 x + 1 (c) log x yz (d) log (x2 y 1) y 3 z 1

29 (e) ln xy ( ) 3x 2 (f) ln y 3 (g) ln ((x + y) 2 (x y)) (h) ln x + 2 x(x 1) ( ) 64x 3 (4x + 1) (i) log 6 (2x 1) ( ) 15x(x 1) (j) log 2 (3x + 2)(2 x) 2

30 2. Contract, expressing the answer as a single logarithm: (a) 1 (log x log(x + 1)) 3 (b) ln x + ln(x 1) (c) ln(x + 1) ln x (d) 2 ln x 3 ln y (e) 1 ln(x + y) 2 (f) ln x 2 ln(2x 1) (g) log(x 2 + x 30) log(x 2 36) 3

31 (h) 2 log(x) 4 log(x + 5) log(7x 1) + 3 log(x 1) 3. Simplify the following: (a) log a a 8 (d) 4 log 4 k (b) ln e t (e) e ln 5 (c) log 10 3k log 7t (f) 10 4

32 4. In chemistry, ph is used to measure the acidity or alkalinity of a substance. The ph scale runs from 0 to 14. Substances with a ph less than 7 are considered acidic, and substances with a ph greater than 7 are said to be alkaline. Our bodies, for instance, must maintain a ph close to 7.35 in order for enzymes to work properly. Below are some common substances and their ph levels: Battery acid: 0.8 Human blood: 7.35 Stomach acid: 2.7 Fresh coconut: 7.8 Orange juice: 3.3 Sodium hydroxide (lye): 14 Pure water: 7 (at 25 C) The formula to find the ph level is found using the following formula, where [H + ] is the concentration of hydrogen ion in the solution: ( ) 1 ph = log([h + ]) = log. [H + ] If the concentration of hydrogen ions in a liquid is doubled, what is the effect on ph? 5. Change log 5 (3) to a quotient of natural logarithms. 5

33 Section 6.6: Exponential and Logarithmic Equations 1. Solve the following: (a) 2 x 1 = 2 2x 4 (b) 5 2x = 5 3x+2 (c) 8 x+2 = 16 x+1 (d) e 2x = 25 3x+2 (e) 2 5x = 2 (f) 3 x+1 = 2 1

34 2. Solve the following: (a) 5 x+2 = 4 x (b) 2 x = 3 x+1 (c) 100 = 20e 2x (d) 3e 0.5x = 11 2

35 (e) 4e 2x + 5 = 12 (f) 3 + e 2x = 7e 2x (g) e 2x = e x + 2 (h) e 2z e z = 56 3

36 3. (a) log 2 (2) + log 2 (3x 5) = 3 (b) 2 ln(x) + 3 = 7 (c) 6 + ln x = 10 (d) 2 ln(x + 1) = 10 (e) log(3x 2) log(2) = log(x + 4) (f) ln(x 2 ) = ln(2x + 3) 4

37 (g) ln x 2 = ln 1 4. If cobalt-60 has a half-life of 5.3 years, how long will it take for 10 percent of 1000 grams of cobalt-60 to decay? 5. The population of a small town is modeled by the equation P = 1650e 0.5t where t is measured in years. In approximately how many years will the town s population reach 20,000? 6. An account with an initial deposit of $6500 earns 7.25% annual interest, compounded continuously. How long will it take the account to double in value? 5