Math Review
1.9 1. y=5√x. (a) Is this a power function? (YES or NO) If it is a power function, answer the following two questions; otherwise leave them blank: Power functions can be written as y=kx^p. For this function, (b) What is k? (c) What is p?
a) Yes b) 5 c) .5
1.8 2. Let h(x) =3x2+4x+1. Let g(x) =3√x+1. (a)g(h(x)) = (b)h(g(x)) = (c)h(h(x)) = (d)g(x)+1= (e)g(x+1) =
a)(3*xˆ2+4*x+1)**(1/3)+1 b)3*(xˆ(1/3)+1)ˆ2+4*(xˆ(1/3)+1)+1 c)3*(3*xˆ2+4*x+1)ˆ2+4*(3*xˆ2+4*x+1)+1 d)xˆ(1/3) + 1 + 1 e)(x+1)ˆ(1/3) + 1
1.7 pt. 2 10. A a population shrinks from its initial level of 16,000 at a continuous decay rate of 5.5% per year. (a) Find a formula for P(t), the population in t years. P(t) =_______ (b) By what percent does the population shrink each year? _______%(Round to the nearest 0.001%)
a)16000e^(-.055t) b)16000e^(-.055(1)) =15143.76 16000-15143.76 =856.24 856.24/16000 =5.351%
1.6 3. Assume the population of a city is 2500 and is growing at a rate of 2% a year. (a) Find a formula for the population of the city at time t years from now assuming that the 2% per year is an annual rate: P=______ For this case, estimate the population of the city in 24 years: _______ (Round off to the nearest whole person.) (b) Find a formula for the population of the city at time t years from now assuming that the 2% per year is a continuous rate: P=____ For this case, estimate the population of the city in 24 years: _______ (Round off to the nearest whole person.)
1. a) 2500(1.02)^t b) 2500(1.02)^24 4021 2. a) 2500e^(.02t) b) 2500e^(.02*24) 4040
1.8 1. a. Write an equation for a graph obtained by vertically stretching the graph of y=x2−7.9 by a factor of 4.5, followed by a vertical shift downward by 49 units. Answer:y=________. b. Write an equation for a graph obtained by vertically shifting the graph of y=x2−7.9 downward by 49 units, followed by stretching the resulting graph by a factor of 4.5. Answer:y=________.
a)4.5(x^(2)-7.9)-49 b)4.5(x^(2)-7.9-49)
1.7 pt. 2 1. An innovative rural public health program is reducing infant mortality in a certain West African country.Pretend the program in Senegal has been reducing infant mortality at a rate 8 % per year. How long willit take for infant mortality to be reduced by 40 %? Answer:_______ years
60/100=100(.92)^t tln92=ln.6 t=ln.6/ln.92 6.1264
1.7 pt. 2 5. Suppose strontium-90 decays at a rate of 2 percent per year. (a) Write the fractionPof strontium remaining, as function oft, measured in years. (Assume that at timet=0 there is 100 % remaining.) Answer:P(t)=______ (b) Estimate the half-life of strontium. Answer:_______ (c) If presently there is 4 grams of strontium, estimate how many grams of the substance will remain after27 years. Answer:_______
a)100(.98)^t b)ln.5/ln.98 c)4(.98)^27 2.32
1.7 pt.1 2. Assume you invest $ 7600 in an account that pays an annual interest rate of 4.6 % that will be compounded continuously. (a) How much money is in the account after 12 years with the given rate? $_______ (b) If you want the account to contain $ 9100 after 12 years, what yearly interest rate is needed? ________%
a)7600e^(.046*12) $13199.09 b)9100=7600e^r(12) 91/76=e^(12r) 12rlne=ln(91/76) ^lne=1 r=(ln(91/76))/12 1.5%
1.7 pt. 2 6. Assume that a population was 865 when t=0, and 6 years later it became 2289. Assuming the population grows exponentially, write a formula for the size of the population in t years: Population= What is the population when t=8?
a)865((2289/865)^(1/6))^t 865(2289/865)^(t/6) b)865(2289/865)^(t/6) 3166.08
1.6 7. If the equation P=16(2.5)^t is rewritten in the form P=P0e^(kt), then: What is P0? What is k?
a. 16 b.ln2.5 .92
1.5 2. A town has 1600 people initially. In each of the cases below, find the formula for the population of the town, P, in terms of the number of years, t. (a) The town grows by 100 people a year. Answer: P(t)= (b) The town grows at an annual rate of 3 percent a year. Answer: P(t)=
a. 1600+100t b. 1600(1.03)^t
1.2 1. Find the equation of the straight line passing through the points(x1,y1) = (−4,2)and(x2,y2) = (7,−10). Answer: y=
y-2=12/11(x+4) y=-12/11x+((-48/11)+2)
1.7 pt.1 3. Suppose that the annual interest rate on your checking account is 8 percent compounded continuously. In order to have $ 9600 in 8 years, how much should you deposit now? Assume, that the current balance is$ 0. Answer: $________
9600=Poe^(.08*8) Po=9600/e^.64 $5062
1.7 pt. 2 3. Assume a quantity is increasing by 9.3% per year. What is the doubling time?_______ years
P=Po(1.093)^t ln2/ln1.093 7.8
1.8 3. Letf(x) =6x+6. Letg(x) =ln(x)+1. Find the following: (a)g(f(x)) = (b)f(g(x)) = (c)f(f(x)) =
a)ln(6x+6)+1 b)6(lnx+1)+6 6lnx+12 c)6(6x+6)+6 36x+36+6
1.6 4. 7^t=10 t=________
ln(A^p)=p(ln*A) ln(7^t)=ln(10) tln7=ln10 t=ln10/ln7 1.18
1.4 pt.2 2. The demand and supply curves for a product are given by q=1400−30p and q=40p−1300, respectively, where p is the price and q is the quantity of the product. (a) Find the equilibrium price and quantity. p= q= (b) A specific tax of $6 is imposed on suppliers. Find the new equilibrium price and quantity. p= q= (c) At the new equilibrium price and quantity, how much of the $6 tax is paid by consumers, and how much by producers? By consumers: $ By producers: $ (d) What is the total tax revenue received by the government? Tax revenue = $
1. a) 2700/70 38.57 b) 40(270/7)-1300 10800/7-1300 242.86 2. a) (2700+(40*6))/70 42 b) 40(p-6)-1300 40p-1540 1400-30p=40p-1540 7p=294 294/7 42 40(42)-1540 1680-1540 140 3. a) 6-2.57 3.43 (do part b first) b) 42-38.57 6-(42-38.57) 2.57 4. 6*140 840
1.9 4. (a)y= (28)/√x is y=kx^p, where k= and p= (b)y= (5√x)^3 is y=kx^p, where k= and p= (c)y= (3/x^2)^−2.5 is y=kx^p, where k= and p=
1. a) 28 b) -.5 2. a) 5^3 125 b) 3/2 1.5 3. a) 3^-2.5 .064 b) 2*2.5 5
The demand and supply curves for a product are given by q=2000−10p and q=40p−1800, respectively, where p is the price and q is the quantity of the product. (a) Find the equilibrium price and quantity. p= q= (b) A sales tax of 7% is imposed on consumers. Find the new equilibrium price and quantity. p= q= (c) How much tax is paid on each unit? Tax per unit: $ (d) At the new equilibrium price and quantity, how much of the tax is paid by consumers, and how much by producers? By consumers: $ By producers: $ (e) What is the total tax revenue received by the government? Tax revenue = $
1. a) 3800/50 76 b) 2000-10(76) 1240 2. a) 3800/(1.07*50) 71.03 b) 2000-10(1.07p)=40p-1800 2000-10.7p=40p-1800 3800=50.7p p=3800/50.7 p=74.95 2000-10.7(74.95) 1198.04 3. .07(74.95) 5.25 4. a) 5.25-1.05 4.19 (do part b first) b) 76-74.95 1.05 5. (.07*74.95)*(2000-10.7(3800/50.7) 6285.51
1.5 4. (1)A=52(0.99)t (1.a.) What is the initial amount (att=0)? (1.b.) Is it exponential growth or decay? (Enter G or D only.) (1.c.) What is the percent growth or decay rate?%
1. a. 52 b. D c. 1
1.7 pt.1 4. Assume a savings account had $5000 initially, and that the interest rate per year to be compounded continuously is 6.4%, and the target balance is $14000. How long will it take to reach the target balance? ________ years
14000=5000e^(.064t) 14/5=e^(.064t) .064tlne=ln7.8 ^=1 t=ln2.8/.064 16.09
1.5 6. Find a possible formula for the exponential function represented by the data in the following table. x 0 1 2 3 f(x). 2.7. 3.78. 5.292. 7.4088 Answer:f(x) =
2.7(1.4)^x
1.7 pt. 2 2. A culture of bacteria grows exponentially. It doubles in size every 94 hours. How long will it take to triple from its original size? Answer:______ hours.
3Po=Po(2^(t/94)) 94ln3=(t/94)ln2 t=(94ln3)/ln2 148.99
1.9 6. Suppose A is inversely proportional to the square of B, and A=5 when B=4. (1) Find the constant of proportionality. ______ help (numbers) (2) Write the formula for A in terms ofB. ______ help (equations) (3) Use your formula to find A when B=7. A=_______ help (numbers)
A=k(1/B^2) 5=k(1/16) k=80 a)80 b)A=80/B^2 A=80B^(-2) c)A=80/7^2 A=80/49 80/(7^2)
1.7 pt.1 1. Determine which is worth more: a) 1125 dollars invested at 10% annual interest rate (compounded yearly) or b) 1600 invested at 7% annual interest (compounded yearly) after I) 4 years? Answer___(choose a or b) II) 30 years? Answer___(choose a or b) III) When would the two investments have equal value? Answer___
P(future value)=Po(present value)*(1+r)(interest rate)^t(time) Pa=1125(1.1)^t Pb=1600(1.07)^t a) b b) a c) Pa(t)=1125(1.1)^(t)=1600(1.07)^(t)=Pb^(t) ln1125+ln(1.1)^(t)=ln1600+ln(1.07)^(t) t(ln1.1/1.07)=ln1600-ln1125 t=(ln1600-ln1125)/(ln1.1-ln1.07) 12.74
1.9 7. The circulation time of a mammal (that is, the average time it takes for all the blood in the body to circulate once and return to the heart) is proportional to the fourth root of the body mass of the mammal.Let the circulation time be Y and the body mass be x. (a) If an elephant of body mass 4200 kilograms has a circulation time of 156 seconds, find the constant of proportionality. Answer (b) Write a formula for the circulation time in terms of body mass. Answer:Y(x) = (c) Assume that the constant of proportionality is the same for all mammals. What is the circulation time of a human with body mass 74 kilograms? Answer:
Y=k(4√x) 156=k4√4200 a)k=156/4√4200 19.38 b)19.38(x^(.25)) c)Y=156/4√4200(4√74) 156*4√74/4200 56.84
1.7 pt. 2 8. Assume a substance has a half-life of 17 years and the initial amount is 101 grams. How much remains at the end of 9 years?______ grams How long will it be until only 40 % remains?______ years
a) 101(.5)^t/17 101(.5)^9/17 69.98 b) .4(101)=101(.5)^(t/17) ln.4=(t/17)ln.5 17ln.4=tln.5 t=(17ln.4)/ln.5 22.47
1.4 pt. 2 1. The tables given below represent a supply curve and a demand curve, not necessarily in that order. Prices are given in dollars and quantity is given in number of units. Table A p 182 164 156 144 131 127 116 q 5. 10 15 20. 25 30 35 Table B p 12 40 61 118 163 242 301 q 5. 10 15 20 25. 30. 35 Answer the following questions. (a) The Supply curve is represented in table _____ (b) Consumers would purchase 25 items of a product at a price of ______ dollars. (c) At what price are manufacturers willing to supply 25 items? Answer ______ dollars.
a) B b) 131 c) 163
1.9 2. y=13/x^9. (a) Is this a power function? (YES or NO)If it is a power function, answer the following two questions; otherwise leave them blank: Power functions can be written as y=kx^p. For this function, (b) What is k? (c) What is p?
a) Yes b) 13 c) -9
1.9 3. y=(19x^3)^4. (a) Is this a power function? (YES or NO)If it is a power function, answer the following two questions; otherwise leave them blank: Power functions can be written as y=kx^p. For this function, (b) What is k? (c) What is p?
a) Yes b) 19^4 130321 c) 3*4 12
1.2 2. Suppose you have a budget of 300 dollars for one month. You would like to buy some textbooks and CDs.The average cost of a book is 50 dollars each and that of a CD is 16 dollars each. Let x denote the number of books you buy and y denote the number of CDs that you buy. (a) Write the equation of your budget constraint in the form y=mx+b, where m and b are numbers predetermined. Answer: y=__________ (b) If you decide to buy just one textbook, how many CDs can you afford? Enter the number of CDs as a whole number. Answer: I can buy ______ CDs.
a. 16y=-50x+30 y=-50/16x+300/16 b. y=-50/16+300/16 y=250/16 y=15.625 15
1.4 1. A company has cost function C(q) =4400+40qdollars and revenue function R(q) =90qdollars where q is the production level. Answer the following questions: (a) What are the fixed costs for the company? Answer: fixed costs=_______ dollars (b) What is the variable cost per unit? Answer: variable cost per unit=_______ dollars (c) What price is the company charging for its product? Answer: price=________ dollars (d) Find the break-even point. Answer: q=_______ units
a. 4400 b. 40 c. 90 d. 4400+40q=90q 50q=4400q 5q=440q q=88
1.4 5. A company that makes jigsaw puzzles has fixed costs of $5350, variable costs of $1 per puzzle, and sells the puzzles for $6 each. For the next three questions complete the given equations. Let x denote the number of puzzles. (a) The Cost function is C(x) = (b) The Revenue function is R(x) = (c) The Profit function is π(x) = (d) Find the break-even point either algebraically or graphically (by sketching the graph of the Revenue and Cost functions using your graphing calculator, and, from the graph, reading the break-even point):x=
a. 5350+x b. 6x c. 6x-(5350+x) 5x-5350 d. 5x-5350=0 5x=5350 x=1070
1.6 9. Solve the following equations for x using the natural logarithm A) 3^x=6 Answer: B) 3e^x=7 Answer: C) 6·7^x=6.5e^−2x Answer:
a. ln(3^x)=ln(6) xln3=ln6 x=ln6/ln3 1.63 b. ln(3e^x)=ln7 ln3+xlne=ln7 ^ xlne=x x=ln7-ln3 .85 c. ln(6*7^x)=ln(6.5e^(-2x)) ln6+xln7=ln6.5+(-2x)lne xln7+2x=ln6.5-ln6 x(ln7+2)=ln6.5-ln6 x=(ln6.5-ln6)/(2+ln7) .02
1.7 pt. 2 9. Pretend the world's population in 1986 was 4.9 billion and that the projection for 2020, assuming exponential growth, is 7 billion. What annual rate of growth is assumed in this prediction?______ % per year
a=(7/4.9)^(1/2020-1986)(7/4.9)^(1/34) (7/4.9)^(1/34)-1 1.06
1.6 6. 19e^(10t)=12e^(9t) t=_______
ln(19e^(10t))=ln(12e^9t) ln19+lne^(10t)=ln12+lne^(9t) 10t(lne)^(1)-(9t)lne^(1)=ln12-ln19 t=ln12-ln19 t=ln(12/19)
1.6 5. 14=10e^(.7t) t=________
ln(AB)=lnA+lnB ln14=ln(10e^.7t) ln14=ln10+.7tlne .7tlne=ln14-ln10 t=(ln14-ln10)/.7 .48
1.7 pt. 2 7. Assume a substance is reduced by 25% in 35 hours. What is the substance's half-life?_____ hours
ln.5/ln.75^(1/35) ln.5/(1/35)ln.75 35ln.5/ln.75 84.33
