Praxis Math Content Knowledge

P(A or B) =

P(A) + P(B) - P(A and B)

Theoretical Probability Formula

P(A) = # of outcomes in event A / total # of outcomes OR P(A) = favorable/total

lim x->0 sinax/sinbx

a/b

If we have a row of all zeros in a matrix, the system is...

dependent

When finding the number of ways that event A OR event B occur...

we add

cos45°

√2/2

sin45°

√2/2

tan60°

√3

cos30°

√3/2

sin60°

√3/2

tan30°

√3/3

Last year Sally spent $5,500 on rent. On a circle graph of her total expenses last year, the sector representing rent has a central angle of 150 degrees. Which of the following is closest to Sally's total expenses last year? -$13,200 -$15,200 -$17,200 -$19,200 -$24,200

$13,200 Solving 360/150 = x/5,500 for x gives us $13,200

Percent error

( |experimental value-accepted value| /accepted value) x 100

Every morning, one student in a class of 24 students is randomly chosen to take attendance. What is the probability that the same student will be chosen 3 days in a row?

(1/24)^3 = 1/13,824

Simplify and express (2.6 x 10^5)(9.2 x 10^-13) in scientific notation.

(2.6 x 10^5)(9.2 x 10^-13) = (2.6)(10^5)(9.2)(10^-13) =(2.6)(9.2)(10^5 * 10^-13) =(2.6)(9.2)(10^-8) =(23.92)(10^-8) =(2.392 * 10^1)(10^-8) =(2.392)(10^1 * 10^-8) =2.392 x 10^-7

Convert 24 kilometers per minute to meters per second

(24 km)/(1 min) x (1,000 m)/(1 km) x (1 min)/(60 secs) = 400 m/s

We need 100 liters of a 25% saline solution and we only have a 14% solution and a 60% solution. How much of each should we mix together to get the 100 liters of the 25% solution?

(Amount of salt in 14% solution) + (Amount of salt in 60% solution) = (Amount of salt in 25% solution) (0.14)(Volume of 14% solution) + (0.6)(Volume of 60% solution) = (0.25)(Volume of 25% solution) .14x + .6(100 - x) = .25(100) .14x + 60 - .6x = 25 -.46x = -35 x = 76.09 liters

Two planes start out 2800 km apart and move towards each other meeting after 3.5 hours. One plane flies at 75 km/hour slower than the other plane. What was the speed of each plane?

(Rate of Plane A)(Time of Plane A) + (Rate of Plane B)(Time of Plane B) = 2800 r(3.5) + (r - 75)(3.5) = 2800 3.5r + 3.5r - 262.5 = 2800 r = 437.5 km/hr

You randomly select 2 cards from a standard deck of 52 cards. What is the probability that the first card is not a heart and the second card is a heart if... (a) you replace the first card before selecting the second card? (b) you do not replace the first card?

(a) P(not heart) * P(heart) = 3/4 * 1/4 = 3/16 (b) P(not heart) * P(heart | not heart) = 3/4 * 13/51 = 13/68

Two rectangular prisms are similar, with one pair of corresponding lengths being 15 cm and 27 cm, respectively. (a) If the volume of the smaller prism is 2000 cm^3, what is the volume of the larger prism? (b) If the area of one face of the larger prism is 243 cm^2, what is the area of the corresponding side of the smaller prism?

(a) V = 11,664 cm^3 (b) A = 75 cm^2 (a) small/large: 15/27 = 5/9 (5/9)^3 = 125/729 125/729 = 2000/V V = (2000)(729)/125 = 11,664 cm^3 (b) (5/9)^2 = 25/81 25/81 = A/243 A = (25)(243)/81 = 75 cm^2

associative property

(a+b)+c=a+(b+c)

Power Rule I for exponents

(a^m)^n = a^m*n

Power Rule II for exponents

(ab)^m = (a^m)(a^n)

standard form of a circle

(x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius

z score formula

(x-mean)/standard deviation

midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2

−16^3/2 =

-(√16)^3 = -(4)^3 = -64

3√-125 =

-5

mean proportional

-Either of the two means of a proportion in which the means are equal. Also called geometric mean -ex: find the mean proportionality of 3 and 12 3/x = x/12 36 = x^2 x = +- 6

Fundamental Counting Principle

-If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is m*n -If three events occur in m, n, and p ways, then the number of ways that all three events can occur is m*n*p

When all outcomes are equally likely, the odds in favor of an event A and the odds against an event A are...

-Odds in favor of event A = number of outcomes in A/number of outcomes NOT in A = favorable/not favorable -Odds against event A = number of outcomes not in A/number of outcomes in A = not favorable/favorable

geometric sequence

-a sequence in which each term is found by multiplying the previous term by the same number -an=a1(r)^n-1

correlation coefficient

-a statistical index of the relationship between two things (from -1 to +1) -strong positive linear relationship has r close to 1 -strong negative linear relationship has r close to -1 -weak relationships have r close to 0

normal distribution (bell curve)

-a symmetrical distribution of data where the mean = median - 68-95-99.7 rule: 68% of the area lies within 1 standard deviation of the mean, 95% of the area lies within 2 standard deviations of the mean, and 99.7% of the area lies within 3 standard deviations of the mean

line of symmetry when graphing in standard form

-b/2a

derivative of cotx

-csc^2x

derivative of -cscx

-cscxcotx

cos(90°)

0

lim x-> (1-cos(ax))/bx

0

lim x->0 (1-cosx)/x

0

sin(0)

0

tan0°

0

cos(0°)

1

sin(90°)

1

tan45°

1

logb(b) =

1 because b^1 = b

finding standard deviation

1. Find the mean 2. For each number, subtract the mean and square the result 3. Find the mean of the squared differences 4. Take the square root

Find the focus equation of the ellipse given by 4x^2 + 9y^2 - 48x + 72y + 144 = 0.

1.) Rewrite the equation: 4x^2 - 48x + 9y^2 + 72y = -144 2.) Factor out the x^2 and y^2 factors: 4(x^2 - 12x) + 9(y^2 + 8y) = -144 3.) Divide by the x factor: (x^2 - 12x) + (9/4)(y^2 + 8y) = -36 4.) Divide by the y factor: (1/9)(x^2 - 12x) + (1/4)(y^2 + 8y) = -4 5.) Complete the square for both groups and add these factors to the right side: (x^2 - 12x + 36)/9 + (y^2 + 8y + 16)/4 = -4 + (36/9) + (16/4) 6.) Factor: ((x - 6)^2)/9 + ((y + 4)^2)/4 = 4 7.) Divide by the right side to get it to equal 1: ((x - 6)^2)/36 + ((y + 4)^2)/16 = 1

Three matrix row operations

1.) switching the rows 2.) scalar multiplication/row multiplication 3.) adding and subtracting rows

cos60°

1/2

sin30°

1/2

A pizza parlor offers 10 different toppings. If no topping is used more than once, how many different 3 topping pizzas can be formed with the 10 toppings?

10 nCr 3 = 120 pizzas

How many different 7 digit telephone numbers are possible if all of the digits can be repeated?

10^7 = 10,000,000

You randomly pick a card from a standard 52 card deck. What are the odds in favor of choosing a face card?

12/40 = 3/10

radians to degrees conversion

180/π

How many different 6 letter arrangements of the letters in the word SPREAD begin with R and end with S or begin with S and end with R?

2 nPr 2 * 4 nPr 4 = 48

We want to fence in a field whose length is twice the width and we have 80 feet of fencing material. If we use all the fencing material what would the dimensions of the field be?

2(Length of Fence) + 2(Width of Fence) = 80 2(x) + 2(2x) = 80 6x = 80 x = 13.33

You are installing rain gutters across the back of your house. The directions say that the gutters should decline 1/4​ inch for every four feet of lateral run. The gutters will be spanning thirty-seven feet. How much lower than the starting point (that is, how much lower than the high end) should the low end of the gutters be?

2.3125 in declination, in/length, ft = (1/4)/4 = d/37 37(1/4) = 4d d = (37/4)/4 d = 9.25/4 = 2.3125 in

formula for period of sine/cosine functions

2π/|B| Standard form: Asin(Bx - C) + D

period of y = asin(bx), y = acsc(bx), y = asec(bx) or y = acos(bx)

2π/|b|

Circumference of a Circle

2πr

To raise funds for a special project, a school sold 999 lottery tickets, numbered consecutively 1 through 999, with a limit of one ticket per person. The school announced that anyone holding a ticket with a number divisible by 45 would win a hat and anyone holding a ticket with a number divisible by 54 would win a T-shirt. How many ticket holders won both a hat and a T-shirt? -0 -3 -9 -18 -22

3 To solve this question, we need to determine how many numbers between 1 and 999 are divisible by both 45 and 54. Such numbers are common multiples of 45 and 54. Let's first write out the prime factorization of 45 and 54. 45=3×3×5 54=2×3×3×3​ Since the unique prime factors are 2, 3, 3, 3, 5, the lcm is 2x3x3x3x5 = 270 Since 45 and 54 evenly divide into 270, they also divide evenly into 270x2 = 540 and 270x3 = 810. Therefore, there are 3 numbers between 1 and 999 that are divisible by both 45 and 54. This means 3 ticket holders own both a hat and a t-shirt.

π/6 radians

30°

A manager can identify employee theft by checking samples of shipments. Among 36 employees, 2 are stealing. If the manager checks on 4 different randomly selected employees, find the probability that neither of the thieves will be identified.

34/36 * 33/35 * 32/34 * 31/33 = 248/315

π/4 radians

45°

You randomly pick a card from a standard 52 card deck. What are the odds against choosing a ten?

48/4 = 12

Solve 4^x = (1/2)^(x - 3)

4^x = (1/2)^(x - 3) (2^2)^x = (2^-1)(x - 3) 2x = -x + 3 3x = 3 x = 1

A circular game board is divided into 16 equal-sized sectors. If the circumference of the game board is 16π inches, what is the area of each sector, in square inches? -2π -4π -8π -16π -64π

4π radius of circle = 8 Area of circle = πr² = π(8)^2 = 64π 64π/16 = 4π

4√x^7 * y^20 * z^11 =

4√x^4 * (y^5)^4 * (z^2)^4 * x^3 * z^3 = 4√x^4 * (y^5)^4 * (z^2)^4 * 4√x^3 * z^3 = x * y^5 * z^2 * 4√x^3 * z^3

If 30 holiday cards can be printed in 2/3 of an hour, how many hours will it take to print 240 cards at this same rate? -4 2/3 -5 1/3 -8 -9 2/3 -12

5 1/3 The printing rate is 45 cards per hour. 240/45 = 5 1/3 hours

How many feet are in a mile?

5,280 feet

Tom is choosing 2 books off of his bookshelf that contains 12 mysteries, 10 comic books, and 5 science fiction novels. He chooses a book and then puts it back on the shelf before picking his next book. What is the probability that he chooses a science fiction book and a mystery?

5/27 * 12/27 = 20/243

π/3 radians

60°

Solve 64^x = 16^(x + 1)

64^x = 16^(x + 1) (4^3)^x = (4^2)^(x + 1) 3x = 2x + 2 x = 2

Find the number of distinguishable permutations of the letters in the word TOMORROW

8!/(3!*2!) = 40,320/12 = 3,360 permutations

π/2 radians

90°

lim x->0 sinx/x

=1

Rational number

A number that can be written as a fraction

arithmetic sequence

A sequence in which the difference between any two consecutive terms is the same

identity matrix

A square matrix with ones (1s) along the main diagonal, from the upper left element to the lower right element, and zeros (0s) everywhere else. ex for 2x2 matrix: 1 0 0 1

One (1) meter is equivalent to which of the following? A. 39.37 inches B. 3.4808333 feet C. 0.914 yards D. All of the above

A. 39.37 inches One meter is equivalent to 39.37 inches. It is also equivalent to 3.2808333 feet and 1.094 yards.

Compound Interest Formula

A=P(1+r/n)^nt where P = principal, r = annual rate (expressed as a decimal), n = number of times compounded per year, A = amount in account after t years

continuously compounded interest formula

A=Pe^rt where P = principal, r = interest rate (as a decimal), A = amount in the account after t years

Area of a circle

A=πr²

Find the product of AB for the following matrices: A = [1 0 -2] [0 3 -1] B = [0 3] [-2 -1] [0. 4]

AB = [((1*0) + (0*-2) + (-2*0)) ((1*3) + (0*-1) + (-2*4))] [((0*0) + (-2*3) + (0*-1)) ((0*3) + (3*-1) + (-1*4))] = [(0 + 0 + 0) (3 + 0 - 8)] [(0 - 6 + 0) (0 - 3 - 4)] = [0 -5] [-6 -7]

How much of a 20% acid solution should we add to 20 gallons of a 42% acid solution to get a 35% acid solution?

Amount of acid in 20% solution + Amount of acid in 42% solution = Amount of acid in 35% solution .2(Volume of 20% solution) + .42(Volume of 42% solution) = .35(Volume of 35% solution) .2x + (.42)(20) = .35(x + 20) .2x + 8.4 = .35x + 7 .15x = 1.4 x = 9.33 gallons

You have made a rectangular quilt that is 5 feet by 4 feet. You want to use the remaining 10 square feet of fabric to add a decorative border of uniform width to the quilt. What should the width of the quilt's border be?

Area quilt + area border = area quilt and border x = width of border (5 + 2x)(4 + 2x) = 20 + 10 4x^2 + 18x + 20 = 30 4x^2 + 18x - 10 = 0 2(2x^2 + 9x - 5) = 0 2(2x - 1)(x + 5) = 0 x = 1/2 or -5 Since x is a length, it cannot equal -5. Therefore, our answer is x = 1/2 foot.

At how many points in the xy-plane do the graphs of y = .25x^4 + .4x^3 - 1.2x^2 -.75x -.25 and y = .5x - 2 intersect? A. 1 B. 2 C. 3 D. 4

B. 2 Use a graphing calculator

A 6 sided number cube is weighted so that the probabilities of throwing 2, 3, 4, 5, or 6 are equal, and the probability of throwing a 1 is twice the probability of throwing a 2. If the number cube is thrown twice what is the probability that the sum of the numbers thrown will be 4? A. 1/12 B. 5/49 C. 3/7 D. 5/11

B. 5/49

Karen opened a bank account for her son on his 1st birthday with a $100 deposit. After that, $50 was deposited into the account on each birthday. No withdrawals and no other deposits were made until his 11th birthday. The bank pays 8% interest per year, compounded annually. Which of the following recursive sequences models the amount of money in the account after n years, 1 <= n <= 10? A.) A(0) = 100 A(n) = .08A(n - 1) + 50 B.) A(0) = 100 A(n) = 1.08A(n - 1) + 50 C.) A(0) = 100 A(n) = .08[A(n - 1) + 50] D.) A(0) = 0 A(n) = 1.08[A(n - 1) + 100] + 50

B. A(0) = 100 A(n) = 1.08A(n - 1) + 50 The initial amount deposited was $100, so A(0) is 100. Every year after the first, the account will gain money from the interest credited plus an additional $50 deposited each year. So, the amount in year n, A(n), will be the amount in year n - 1, 0.08A(n - 1) PLUS an additional $50 deposit (there is no interest on the $50, since it was just deposited and was not in the account all year). The situation can be modeled by the recursive equation A(n) = A(n - 1) + 0.08A(n - 1) + 50 = 1.08A(n - 1) + 50.

The Richter scale is a base 10 logarithmic scale used to measure the magnitude of earthquakes; i.e., an earthquake measuring 7 is 10 times as strong as an earthquake measuring 6. An earthquake that measures 6.8 on the Richter scale has a magnitude that is approximately what percent of that of an earthquake measuring 6.6? A. 103% B. 120% C. 158% D. 200%

C. 158%

If y = 2x + 1, what is the arithmetic mean of 2x, 2x, y, and 3y, in terms of x? A. 2x B. 2x + 1 C. 3x + 1 D. None of the above

C. 3x + 1 The arithmetic mean (average) is equal to the sum of the items divided by the number of items. Average = (2x + 2x + y + 3y) ÷ 4 = (4x + 4y) ÷ 4 = x + y Substituting 2x + 1 for y: x + (2x + 1) = 3x + 1 Check (assume x = 3): (2(3) + 2(3) + 7 + 3(7)) ÷ 4 = (6 + 6 + 7 + 21) ÷ 4 = 40 ÷ 4 = 10 = 3(3) + 1

When solving an equation with real coefficients, which of the following procedures can result in an equation that yields a real root that does not satisfy the original equation? A. Subtracting the same number from both sides of the equation B. Raising both sides of the equation to the third power C. Squaring both sides of the equation D. Dividing both sides of the equation by a nonzero number

C. Squaring both sides of the equation

The measurement of a quantity has an error of p percent if p/100 = |a - m|/a, where a is the actual value and m is the measured value. The measured value of the surface area of a lake is 10200 square meters, and the measurement of error is at most 5 percent. Which of the following could be the actual surface area of the lake, in square meters? I. 9500 II. 10000 III. 10500 A. None B. II only C. I and II D. II and III

D. II and III

If 3/7 is expressed in decimal form, what is the digit in the 19th decimal place? A.) 1 B.) 2 C.) 3 D.) 4

D.) 4

Mike starts out 35 feet in front of Kim and they both start moving towards the right at the same time. Mike moves at 2 ft/sec while Kim moves at 3.4 ft/sec. How long will it take for Kim to catch up with Mike?

Distance Kim moved = 35 + Distance Mike moved (Rate of Kim)(Time of Kim) = 35 + (Rate of Mike)(Time of Mike) 3.4t = 35 + 2t 1.4t = 35 t = 25 s

9/(3√2x) =

For this problem we need to multiply the numerator and denominator by 3√(2x)^2 in order to rationalize the denominator. 9/(3√2x) * (3√(2x)^2)/(3√(2x)^2) =9(3√(2x)^2)/(3√(2x)^3) =9(3√(2x)^2)/2x =9(3√4x^2)/2x

Determinant of a 3 x 3 matrix

If A = [a b c] [d e f] [g h i] then |A| = a(ei - fh) - b(di - fg) + c(dh - eg) The step-by-step instructions are: 1.) Multiply a by the determinant of the 2x2 matrix that is not in a's row or column 2.) Likewise for both b and c 3.) Sum them up but remember the minus in front of the b

Determinant for a 2 x 2 matrix

If A = [a b] [c d] then |A| = ad - bc

Definition of Logarithmic with Base b

Let b and y be positive numbers with b not equal to 1. The logarithm of y with base b is denoted by logb(y) is defined as follows: logb(y) = x if and only if b^x = y.

joint variation

Occurs when a quantity varies directly with the product of two or more quantities z = axy

The average (arithmetic mean) of five different positive integers is 62. If three of the integers are 73, 62, and 45, what is the maximum possible value of the largest of the five integers? A. 73 B. 74 C. 129 D. 130

Option (C) is correct. Assume that the five different positive integers are 73, 62, 45, x, and y, where x and y are positive integers and x<yx<y. Since the average of the five integers is 62, the sum of the five integers is 62×562×5, or 310. The equation 73+62+45+x+y=310 73+62+45+x+y=310 is equivalent to x+y=130x+y=130. Since x and y are positive integers and x<yx<y, the largest possible value of y is 129, which corresponds to an x value of 1. It follows that the sum of 73, 62, 45, 1, and 129 satisfy all the conditions of the problem. Therefore, the correct answer is (C).

Probability of Independent Events

P(A and B) = P(A) * P(B)

Probability of Dependent Events

P(A and B) = P(A) * P(B|A)

Complement of event A formula

P(A) + P(A^c) = 1

Probability of A or B (A and B are disjoint)

P(A) + P(B)

John can paint a house in 28 hours. John and Dave can paint the house in 17 hours working together. How long would it take Dave to paint the house by himself?

Rate of John: r(28) = 1 -> r = 1/28 Rate of Dave: r(t) = 1 -> r = 1/t Portion done by John + Portion done by Dave = 1 job 1/28(17) + (1/t)(17) = 1 17/28 + 17/t = 1 1 - 17/28 = 17/t 11/28 = 17/t t = 43.27 hours

A pump can empty a pool in 7 hours and a different pump can empty the same pool in 12 hours. How long does it take for both pumps working together to empty the pool?

Rate of first pump: r(7) = 1 -> r = 1/7 Rate of second pump: r(12) = 1 -> r = 1/12 (Portion of job done by first pump) + (portion of job done by second pump) = 1 job (Rate of first pump)(Time of first pump) + (Rate of second pump)(Time of second pump) = 1 1/7t + 1/12t = 1 18/84t = 1 t = 4.42 hours

Write in standard form: (7−i)/(2+10i) =

Rationalize denominator: (7−i)/(2+10i) * (2−10i)/(2−10i) =(7−i)(2−10i)/(2+10i)(2−10i) = (14 - 72i + 10i^2)/(4 - 100i^2) =(14 - 72i -10)/(4 + 100) =(4 - 72i)/104 =1/26 - (9/13)i

If n is divided by 5, the remainder is 3. What is the remainder when 3n is divided by 5?

Since when n is divided by 5, the remainder is 3, n = 5k + 3, where k is some whole integer. Therefore, 3n = 15k + 9 ⇒ when 15k + 9 is divided by 5, since 15k is divisible by 5, the remainder is 4 when 9 is divided by 5.

end behavior

The behavior of the graph as x approaches positive infinity or negative infinity.

The width of a rectangle is 8 inches and remains constant, while the length of the rectangle increases at a constant rate of 3 inches per minute. When the length of the rectangle is 6 inches, what is the rate of change, in inches per minute, of the length of a diagonal in the rectangle?

The correct answer is 1.8. This is a calculus-related rate problem. To solve this problem, you need to represent the dimensions of the rectangle in an equation that will pertain to the dimensions throughout the time interval of interest, take the first derivative of the equation with respect to time t, substitute the known quantities and rates of change, and then solve for the desired time rate of change. The Pythagorean theorem will properly relate the variables of interest throughout the desired time interval. So we start the Pythagorean theorem: t^2 + w^2 = x^2, where l represents the length, w represents the width, and x represents the diagonal of the rectangle. The related rate equation is 2l[dl/dt] + 2w[dw/dt] = 2x[dx/dt] or, dividing by 2, l[dl/dt] + w[dw/dt] = x[dx/dt]. Using the original equation to solve for the diagonal, when the length is 6 inches: 6^2 + 8^2 = x^2, we find that x = 10 when l = 6. Substituting the given quantities, we achieve the new related rate equation: 6(3) + 8(0) = 10[dx/dt]. Solving the equation for dx/dt, we find that the rate of change of the diagonal at the moment of interest is 1.8 inches per minute.

What is the value of lim tan(2x)/3x(cos(4x)) ? x -> 0 Give your answer as a fraction.

The correct answer is 2/3. To evaluate the limit, we can use the ratio identity tanx = sinx/cosx and the facts that lim x->0 sinx/x = 1, lim x->0 cosx = 1, and lim x -> 0 x/x = 1: lim x -> 0 tan(2x)/3x(cos4x) = lim x-> 0 [sin(2x)/cos(2x)]/1 * 1/3x * 1/cos(4x) = lim x-> 0 sin(2x)/cos(2x) * 1/3x * 1/cos(4x) = lim x -> 0 sin(2x)/2x * 2x/3x * 1/cos(2x) * 1/cos(4x) = 1 * 2/3 * 1 * 1 = 2/3 Alternate way to solve: L'Hopital's Rule

If p, q, and r are statements and p is true, then the statement (p ^ q) ---> (p v r) is -true -false -true only if q is true -false only if q is true

True Explanation: Since p is true, p ^ q is equivalent to q, and p v r is true. Thus, (p ^ q) ---> (p v r) is equivalent to q ---> true, which is always true.

You buy a new car for $22,500. The value of the car decreases by 25% each year. Write an exponential decay model giving the car's value V (in dollars) after t years.

V = $22,500(1 - .25)^t

You are designing a marble basin that will hold a foundation for a city park. The basin's sides and bottom should be 1 foot thick. Its outer length should be twice its outer width and outer height. What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?

V = lwh (2x -2)(x - 2)(x - 1) = 36 (2x^2 - 6X + 4)(x - 1) = 36 2x^3 - 8x^2 + 10x - 40 = 0 2(x^3 - 4x^2 + 5x - 20) = 0 2(x^2 + 5)(x - 4) = 0 x = 4 ft

Experimental Probability of an event

When an experiment is performed that consists of a certain number of trials, the experimental probability of an event A is given by: P(A) = number of trials where A occurs/total number of trials

The local movie rental store is having a special on new releases. The new releases consist of 12 comedies, 8 action, 7 drama, 5 suspense, and 9 family movies. You can afford at most 2 movies. How many movie combinations can you rent?

You can rent 0, 1, or 2 movies. Because there are 41 movies to choose from, the number of possible sets of movies is: 41 nCr 0 + 41 nCr 1 + 41 nCr 2 = 1 + 41 + 820 = 862

natural base exponential function

a function of the form f(x) = ae^rx, where e is the base -if a > 0 and r >0, the function is an exponential growth function -if a > 0 and r < 0, the function is an exponential decay function

Irrational number

a number that can NOT be expressed as a ratio of two integers or as a repeating or terminating decimal - Pi or any square root of an imperfect square are considered irrational

consistent system

a system of equations or inequalities that has at least one solution

inconsistent system

a system of equations or inequalities that has no solution

distributive property

a(b + c) = ab + ac

inverse property

a+(-a)=0

identity property

a+0=a or a*1=a

commutative property

a+b=b+a

In the xy-plane, what is the radius of the circle described by the equation x^2 + 2x + y^2 = 0? a.) 1 b.) 2 c.) 3 d.) 4

a.) 1 To determine the radius of a circle, it is useful to put the equation of the circle in standard form (x - a)^2 + (y - b)^2 = r^2, where (a,b) is the center of the circle and r is the radius of the circle. To put the equation into standard form, it is necessary to complete the square: x^2 + 2x + y^2 = 0 (x^2 +2x + 1) + y^2 = 1 (x + 1)^2 + y^2 = 1 Now that the equation is in standard form, it is evident that the circle has its center at (-1, 0) and its radius is 1. Therefore, A is the correct answer.

A community center hosts a talent contest for local musicians. On a given evening, 7 musicians are scheduled to perform. The order in which the musicians perform is randomly selected during the show. a.) What is the probability that the musicians perform in alphabetical order by their last names? (Assume that no two musicians have the same last name.) b.) You are friends with 4 of the musicians. What is the probability that the first 2 performers are your friends?

a.) P(performers in alphabetical order) = alphabetical order/total orders = 1/7! = 1/5040 b.) P(1st 2 performers are friends) = 1st 2 performers/total sets of 2 performers = (4 nCr 2)/(7 nCr 2) = 2/7

Negative exponent

a^-m = 1/a^m

difference of cubes

a^3-b^3=(a-b)(a^2+ab+b^2)

Product Rule for exponents

a^m * a^n = a^m+n

quotient rule for exponents

a^m/a^n = a^m-n

Write an explicit formula for the sequence: .2, .04, .008, .0016,...

an = .2(.2)^n-1

Write an explicit formula for the sequence: 2, 7, 12, 17,...

an = 2 + (n-1)5 an = 2 + 5n - 5 an = 5n - 3

Write an explicit formula for the sequence: 2, 10, 50, 250,...

an = 2(5)^n-1

Write an explicit formula for the sequence: 8, 3, -2, -7,...

an = 8 - 5(n - 1) an = 8 - 5n + 5 an = 13 - 5n

growth factor of an exponential function

b in y = ab^x

A stone is projected with a slingshot vertically upward. The height h, in feet, of the stone t seconds after it is projected is given by the equation h = -16t^2 + 80t + 5. What is the average rate of change, in feet per second, of the height of the stone for the 2-second time interval from t=3 seconds to t=5 seconds? a.) -56.5 b.) -48.0 c.) 52.5 d.) 64.0

b.) -48.0 The average rate of change of the height of the stone, in feet per second, is found by dividing the change in height, in feet, by the change in time, in seconds, for the time interval. The average rate of change, in feet per second, is given by the formula v = [h(5) - h(3)]/5-3 = 5 - 101/2 = -96/2 = -48.

What is the slope of the line in the xy-plane that passes through (0,0) and is tangent to the graph of the curve y = x^3 + 16? a.) 2 b.) 12 c.) 16 d.) 24

b.) 12 Assume that the line is tangent to the graph of the curve y = x^3 + 16 at the point (a, a^3 + 16). Since dy/dx = 3x^2, the slope of the line is 3a^2 and the equation of the line is y - (a^3 + 16) = 3a^2(x - a). Since the line passes through the origin, we can substitute x = 0 and y = 0 into the equation of the line, and so we get the equation 0 - (a^3 + 16) = 3a^2(0 - a). This equation in a has a single solution, a = 2. Thus, the slope of the line is 3a^2 = 12.

For a normally distributed population, approximately 68% of the population lies within 1 standard deviation of the mean and approximately 95% of the population lies within 2 standard deviations of the mean. In a certain environment study, the weights of 10,000 fish are normally distributed with a mean of 12.5 ounces and a standard deviation of 4.2 ounces. Approximately what percent of the 10,000 fish have weights between 16.7 and 20.9 ounces? a.) 7% b.) 14% c.) 27% d.) 34%

b.) 14% For a normally distributed population, the z-score, z = (x - μ)/σ, can be used to normalize a weight x, where μ and σ are the mean and standard deviation of the population, respectively. The z-score of a weight of 16.7 ounces is z = 16.7 - 12.5/4.2 = 1, so a weight of 16.7 ounces is 1 standard deviation above the mean. The z-score of a weight of 20.9 is z = 20.9 - 12.5/4.2 = 2, so a weight of 20.9 ounces is 2 standard deviations above the mean. The fish with weights between 16.7 and 20.9 ounces represent the group that lies between 1 standard deviation and 2 standard deviations above the mean. Since 68% of the population lies within 1 standard deviation of the mean, and 95% of the population lies within 2 standard deviations of the mean, the percent of the population that lies between 1 and 2 standard deviations above the mean is 95% - 68%/2, or 13.5%.

Beginning at time t=0, the number of rabbits in a certain population at time t years is modeled by the function f(t) = 10,000/(10 + 50e^-0.5t). According to this model, which of the following best describes how the size of the population changes over time? a.) It increases without bound. b.) It increases for several years then levels off. c.) It increases for several years and then decreases to zero. d.) It increases and decreases in cycles.

b.) It increases for several years then levels off. In order to see the behavior of the function over time, it is useful to graph the function. Notice that at time t=0, the value of the function is 10,000/(10 + 50), or 166 2/3. So a good viewing window to start with might be [0,20] x [0,1000]. Graphing the function in this viewing window shows that the function grows rapidly at first and then levels off. Zooming shows that the function does stay level.

Discriminant

b^2-4ac

Graph of EVEN degree polynomial, NEGATIVE leading coefficient

both ends move to negative infinity

Graph of EVEN degree polynomial, POSITIVE leading coefficient

both ends move to positive infinity

p(x)/(x - 3) = q(x) + 2/(x - 3) for all values x ≠ 3 In the equation above, p(x) and q(x) are polynomial functions. What is the value of p(3)? a.) -3 b.) -2 c.) 2 d.) undefined

c.) 2 The equation shows that when the polynomial is divided by the monomial x - 3, the quotient is the polynomial q(x) and the remainder is 2. Using the remainder theorem, the remainder when the polynomial p(x) is divided by x - 3 is p(3). Thus, we can conclude that p(3) = 2.

d/dx[∫(from 1 to 3x) (1/1 + u) du] = a.) ln(1 + 3x) b.) 1/(1 + 3x) c.) 3/(1 + 3x) d.) 1/(3 + 9x)

c.) 3/(1 + 3x) We need to remember a fundamental theorem of calculus, that d/dx(∫(from c to x) f(u) du) = f(x), where f(u) is continuous from c to x, and that d/dx(f(v)) = df/dv * dv/dx. Let v = 3x, then: d/dx[∫(from 1 to 3x) (1/1 + u) du] =d/dv[∫(from 1 to v) 1/(1 + u) du] * dv/dx =1/(1 + v) * dv/dx =1/(1 + 3x) * 3 =3/(1 + 3x)

A new temperature scale, T, is designed so that a temperature of 0°T is equivalent to a temperature of -24°C on the Celsius scale. If every increase of 5° on the T scale is equivalent to an increase of 8° on the Celsius scale, what is the temperature on the T scale that is equivalent to a temperature of 80°C? a.) 40°T b.) 48°T c.) 65°T d.) 80°T

c.) 65°T It is given that an increase of 5° on the T scale is equivalent to an increase of 8° on the C scale. This means that there is a linear relationship of the form t = 5/8c + b between temperatures t on the T scale and temperatures c on the C scale. Since t=0 when c = -24, it follows that 0 = 5/8(-24) + b, so b = 15. Thus, the temperature on the T scale equivalent to a temperature of 80°C is t = 5/8(80°) + 15 = 65°T.

If a, b, and c are real numbers such that sqrt(b^2 -4ac) = 0, which of the following statements about the roots of the equation ax^2 + bx + c = 0, where a ≠ 0, must be true? a.) The equation has two distinct real roots. b.) The equation has two distinct nonreal roots. c.) The equation has one real root. d.) The equation has one non real root.

c.) The equation has one real root. Explanation: Since sqrt(b^2 - 4ac) = 0, the 2 solutions coming from the quadratic formula are the same number. Thus, the equation has one real root.

If the degree of the numerator = the degree of the denominator in a rational function...

coefficient of numerator/coefficient of denominator is the horizontal asymptote

cot^2x+1 =

csc^2x

What is the period of the function y = 1/3sin[(1/2)x + π/3]? a.) 1/2 b.) π/6 c.) π d.) 4π

d.) 4π In the equation, the term π/3 represents a phase shift of the function y = 1/3sin((1/2)x), so those functions will have the same period. Now the period of y = 1/3sin((1/2)x) can be determined to be 4π by graphing the function in the viewing window [-4π, 4π] x [-1, 1].

The surface area of a sphere is approximately 1,500 square inches. What is the approximate volume of the sphere, in cubic inches? a.) 500 b.) 1,500 c.) 3,500 d.) 5,500

d.) 5,500 The surface area S of a sphere is given in terms of its radius r by the formula S = 4πr^2. So, a sphere with surface area equal to 1,500 square inches will have a radius r = sqrt(1,500/4π) = around 10.93 inches. For a sphere with a radius r, the volume V of the sphere is given by V = (4/3)πr^3, so for a sphere with radius r = 10.93 inches, the volume of the sphere is given by V = (4/3)π(10.93)^3 = 5,470 cubic inches. Therefore, the correct answer is D.

Water started leaking from a tank at noon and continued leaking at a constant rate for the next 6 hours. The volume of water in the tank at 6 P.M. was 1/4 of the volume of water in the tank at noon. Which of the following equations relates (i) g, the number of gallons of water in the tank at noon (ii) t, the number of hours after noon when 0<t<6, and (iii)h, the number of gallons of water in the tank at t hours after noon? a.) 3h + gt = 3g b.) 4h + gt = 4g c.) 4h + 3gt = 4g d.) 8h + gt = 8g

d.) 8h + gt = 8g The amount of water in the tank at time t can be written as h(t) = g + rt, where r is the flow rate of the water. It is given that after 6 hours, there is 1/4 the original amount; this can be written as h(6) = (1/4)g. Substitution yields (1/4)g = g + 6r and solving for r gives r = (-1/8)g (Note: the rate being negative is appropriate since water is leaking out of the tank). Substituting into the original equation for h, and suppressing the argument of the function, gives h = g - (1/8)gt. Multiplying by 8 and then adding gt to both sides of the equation produces the correct answer of 8h + gt = 8g.

Completing the Square Method

divide b by 2 and square it to find c, and then find the root

Average rate of change

f(b)-f(a)/b-a

hyperbola

f(x)=1/x

odd function

graph is symmetrical with respect to the origin; f(-x)=-f(x)

even function

graph is symmetrical with respect to the y-axis; f(x) = f(-x)

A water balloon is dropped from a window 59 feet above the sidewalk. How long does it take for the water balloon to hit the sidewalk? (Hint: Formula for dropping or throwing an object is h(t) = -16t^2 +vt + h)

h(t) = 0 ft v = 0 ft/s h = 59 ft t = ? -16t^2 + 0t + 59 = 0 -16t^2 = -59 t = 1.92 secs

Accuracy

how close a measurement is to the true value

Precision

how close the measured values are to each other

If the discriminant is positive,

it has two solutions

If the discriminant is 0,

it only has one solution.

A matrix is not invertible if...

its determinant is equal to 0.

Graph of odd degree polynomial, negative leading coefficient

left side moves toward negative infinity, right side moves to positive infinity

Graph of odd degree polynomial, positive leading coefficient

left side moves toward positive infinity, right side moves to negative infinity

logb(a) =

log a/log b

Power Property of Logarithms

log(b)m^n= n log(b)m

log2(16^x) =

log2(16^x) = log2(2^4)^x = 4x

Quotient Property of Logarithms

logb (m/n) = logb m - logb n

product property of logarithms

logb mn = logb m + logb n

change of base formula

logc a=logb a/logb c

George has heard from two different sources about the pay range at a particular company. One source says that the ratio of lowest pay to highest pay is 3 : 7. The other source says that the top earner annually makes about $57,000 more than the lowest earner. What are the approximate salaries for the highest and lowest earners? (Round to the nearest thousand.)

lowest: $43,000 highest: $100,000 lowest/highest = 3/7 = L/L + 57 3(L + 57) = 7L 3L + 171 = 7L 4L = 171 L = 42.75

Volume of a Quadrilateral

lwh

Degrees to radians conversion

multiply by π/180

Permutations with repetition

n!/p!q! where p and q are repetitions

Combinations Formula

nCr = n!/r!(n-r)!

Permutation Formula

nPr = n!/(n-r)! n = number of items r = arrangements

Can an absolute value equal a negative number?

no

Is matrix multiplication commutative?

no

A widget is being sold in a store for $135.40 and has been marked up 7%. How much did the store pay for the widget?

p = price store paid for widget p + .07p = 135.4 1.07p = 135.4 p = $126.54

Distance =

rate x time

The product of two rational numbers is

rational

The sum of two rational numbers is

rational

derivative of tanx

sec^2x

tan^2x+1 =

sec^2x

derivative of secx

secxtanx

Product of two irrational numbers is...

sometimes irrational ex: √2 x √3 = √6 (Irrational) √2 x √2 = √4 = 2 (Rational)

The sum of two irrational numbers is

sometimes irrational ex: √2+√2 = 2√2 is irrational 2+2√5+(-2√5) = 2 is rational

graphing exponential functions

take the function f(x) = ab^x the curve intersects the y-axis at the focal (0,a) NOTES: 1) when a>0 the graph is totally above the x-axis 2) when a<0 the graph is totally below the x-axis 3) when a>0 and b>1 the graph will increase away from the x-axis 4) when a<0 and b<1 the graph will increase away from the x-axis 5) if 0<b<1 the graph decreases towards, but never reaches, the x-axis (the horizontal asymptote) 6) ex: y = [2*3^(x - 2)] - 3 -> this graph shifts to the right 2 bc of the minus 2 and shifts down 3 bc of the minus 3

Recursive Sequence

tells how an is related to one or more preceding terms

If the discriminant is negative,

the answer is no solution.

radicand

the number/expression under the radical sign

Order of magnitude

the order of magnitude of a quantity is the number rounded to the nearest power of 10

If log3(x) = log3(8)....

then x = 8

If the degree of the numerator > the degree of the denominator in a rational function...

there is no horizontal asymptote

You can add or subtract matrices only if...

they have the SAME dimensions

inverse variation

two variables x and y show inverse variation if they are related as follows: y = a/x or a = xy where a is the constant of variation

tan90°

undefined

When finding the number of ways both event A AND event B occur...

we multiply

a^loga(x) =

x

e^lnx =

x

ln e^x =

x

The function f(x) = 6x^4 + x^3 -x^2 + 75x - 25 has four distinct zeros. Two of the zeros are x = -5/2 and x = 1/3. What are the two other zeros?

x = 1 +/- 2i

sum of cubes

x³ + y³ = (x + y)(x² - xy + y²)

When a real life quantity decreases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation...

y - a(1 - r)^t where a is the initial amount and r is the percent rate decrease (as a decimal)

Standard form of a sine function

y = A sin (Bx - C) + D

When a real life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation...

y = a(1 + r)^t where a is the initial amount and r is the percent rate of increase

exponential decay function

y = ab^x if a > 0 and 0 < b < 1

derivative of y = a^f(x)

y' = a^f(x) * f'(x) * ln(a)

If the degree of the numerator < the degree of the denominator in a rational function...

y=0 is the horizontal asymptote

exponential function

y=ab^x

period of y = atan(bx) or y = acot(bx)

π/|b|