Quantitative Reasoning
What does the value of the first derivative of a function tell us about a function?
dy/dx > 0 positive slope; function is increasing dy/dx < 0 negative slope; function is decreasing dy/dx = 0 no slope; function is neither increasing nor decreasing
Given two functions: f(x) = x^2 + x - 12 g(x) = 2x Find f ᵒ g(3) and g ᵒ f(3)
f ᵒ g(3) = f(32x) = (2x)^2 + (2x) - 12 =4x^2+2x-12 =4(3)^2+2(3)-12 = 30 g ᵒ f(3) = g(x^2+x-12) = 2(x^2+x-12) =2x^2+2x-24 =2(3)^2+2(3)-24 = 0
what is the derivative for the following function? f(x) = x^4 - 2x^3 + x^2 + 3x - 4/x -12
f(x) = x^4 - 2x^3 + x^2 + 3x - 4x^-1 -12 f'(x) = 4x^3 - 6x^2 + 2x + 3 + 4x^-2
What does i = ?
i = sqrt(-1)
What is the characteristic shape for each of the following functions? (i) y = 2x+4 (ii) y = |2x-4| (iii) y = x^2+2x-8 (iv) y = sqrt(x+5)
(i) This is the definition of a straight line (ii) This is an absolute value function and will have a V-shape (iii) This is a quadratic function and will be a parabola (iv) This is a radical function and will have a shape that looks like the profile of a frisbee
How do we use deriviatives to determine local maxima and local minima?
1. Find all points where f'(x) = 0 2. Find the value of f''(x) at these points 3. If f''(x) > 0, the point is a local minimum If f''(x) < 0, the point is a local maximum
What are the criteria that must be met for a function to be continuous?
1. For a given function f(x), f(a) must be defined 2. lim(x-->a) f(x) has a finite value 3. lim(x-->a) f(x) = f(a)
Find the following integral: int(x^3- (1/2)x^2 - x + 3)dx
1/4) x^4 - (1/6) x^3 - (1/2) x^2 + 3x + c
sqrt16 = ?
sqrtn is defined as the non-negative number that when squared equals n. There are two numbers that when squared equals 16: 4 and -4. By definition, however, sqrt16 means the non-negative square root: 4.
Identify the real and imaginary parts in the following complex number: 3 + 4i
3 = real +4i = imaginary
7^3 = ?
7^3 = 7*7*7 = 343
Factor: 3x^3 + 12x^2 -6x
A factor common to all terms of a polynomial can be factored out. ALl three terms in the polynomial 3x^3 + 12x^2 - 6x contain a factor of 3x. Pulling out the common factor yields 3x(x^2+4x-2).
What is a limit, and when do functions have a limit
A limit is the number a function approaches. A limit exists when the function approaches the same number from both the left and right sides
(-1)^57 = ?
A negative number raised to an even power yields a positive result. A negative number raised to an odd power yields a negative result. -1 to an even power is 1; -1 to an odd power is -1. Therefore, the answer is -1.
Express 32% as a fraction in lowest terms.
A percent is a fraction with an implied denominator of 100. 32% means 32/100 which reduced to 8/25.
What is the greatest prime number less than 37?
A prime number is a positive integer that is divisible only by 1 and itself. The smallest prime number, and the only even prime number, is 2. The sequence of the first 15 primes is as follows: 2, 3, 5, 7, 11, 13, 17 19, 23, 29, 31, 37, 41, 43, 47. Therefore, the greatest prime number less than 37 is 31.
What is the next number in the following sequence? 39, 42, 45, 48
A sequence is a series of numbers following a particular pattern. The numbers 39, 42, 45, and 48 are consecutive multiples of 3. Each number in the sequence is 3 more than the previous number. The next number would be 48 + 3 = 51
What is the formula for the cross product?
A x B = |A||B|sin(theta) where A and B are two vectors and theta is the angle between them
Which of the following is an integer? -7/2, -0.1, 12%, 3, pi
An integer is a miltiple of 1. Integers include negative whole numbers and zero. The only integer listed is 3.
Which of the following is a multiple of 2 but not a multiple of 4? 124, 353, 484, 562, 708, 984
An integer is divisible by 2 (even) if the last digit is even. An integer is divisible by 4 if the last two digits make a multiple of 4. The last digit of 562 is 2, which is even, so 562 is a multiple of 2. The last two digits make 62, which is not divisible by 4, so 562 is not divisible by 4.
Which of the following is a multiple of 3 but not a multiple of 9? 109, 117, 260, 361, 459, 957, 1001
An integer is divisible by 3 if the sum of its digits is divisible by 3. An integer is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits in 957 is 21, which is divisible by 3 but not by 9, so 957 is divisible by 3 but not by 9.
Alex drove 120 miles at 40 miles per hour and another 120 miles at 60 miles per hour. What was Alex's average speed for the whole 240 miles?
Average A per B = Total A/Total B. Average speed = Total distance/Total time. To find the average speed for 120 miles at 40 mph and 120 miles at 60 mph, don't just average the two speeds. First figure out the total distance and the total time. The total distance is 120+120 = 240 miles. The times are 3 hours for the first leg and 2 hours for the second leg, or 5 hours total. The average speed, then, is 240/5 = 48 miles per hour.
What are the sum and difference of the following complex numbers? z = 3 + 2i w = 1 - 3i
z + w = (3 + 2i) + (1 - 3i) = 4 - i z - w = (3 + 2i) - (1 - 3i) = 2 + 5i
What is the equation for standard deviation?
σ = √∑(x-x̅)^2/n To find the standard deviation, you must first find the mean. Then you sum the squares of the difference of each number and the mean, divide by the number of terms, and take the square root.
How many integers are there from 13 through 31, inclusive?
Consecutive numbers are numbers of a certain type, presented in order without skipping any. To count consecutive integers, subtract the smallest from the largest and add 1. To count the integers from 13 through 31, subtract: 31 - 13 = 18. Then add 1: 18 + 1 = 19
How many distinct digits are in the number 321,321,000
Digits are the symbols used in numerals (combination of symbols) to represent numbers. The number 321,321,000 has 9 digits, but only 4 distinct (different) digits: 3, 2, 1, and 0.
How many hours are needed to travel 220 miles at 55 miles per hour?
Distance = Rate * Time 220 miles = (55 miles per hour)*(hours) 220 = 55x x = 220/55 = 4
What does evaluating a definite integral mean?
Evaluating a definite integral determines the area under the curve between the limits of integration
What do the following prefixes mean? Tera, giga, mega, kilo, hecto, deca, deci, centi, milli, micro, nano, pico
FACTOR/NAME/SYMBOL 10^12/tera/T 10^9/giga/G 10^6/mega/M 10^3/kilo/k 10^2/hecto/h 10^1/deca/da 10^-1/deci/d 10^-2/centi/c 10^-3/milli/m 10^-6/micro/mu 10^-9/nano/n 10^-12/pico/p
What is the limit for the following function? lim(x-->1) (x^2-4)/(x-2)
Here we use the factoring method, followed by substitution. lim(x-->1)(x^2-4)/(x-2) = (x+2)(x-2)/(x-2) = x + 2 = 1 + 2 = 3
What is the limit for the following function? lim(x-->1) (x+2)
Here we use the substitution method: lim(x-->1) (x+2) = 1 + 2 = 3
Find the horizontal asymptotes of the following functions: (i) y = (x+3)/(x^2+2x-15) (ii) y = (2x^2-8)/(x^2+x-12)
Horizontal asymtpoes indicate graph trends usually at a distance from the origin. They occur when a function's numerator grows only as fast as or slower than the denominator. (i) When a denominator's highest degree is larger than the numerator's, the function will have a horizontal asymptote at y = 0. (ii) When the denominator's highest degree is the same as the numerator's, the function will have a horizontal asymptote at y = the ratio of highest degree coefficients, in this case y = 2.
In a group of 18 people, the ratio of men to women is 1:2. How many women are there?
If the parts add up to the whole, a part-to-part ratio can be turned into 2 part-to-whole ratios by putting each number in the original ratio over the sum of the numbers. If the ratio of men to women is 1 to 2, then the men-to-people to ratio is 1/(1+2) = 1/3 and the women-to-people ratio is 2/(1+2) = 2/3. In other words, 2/3 of the 18 people, or 12, are women.
Factor: x^2-9
One of the test maker's favorite factorables is the difference of squares. An expression in the form a^2-b^2 factors to (a-b)(a+b). x^2-9, then, factors to (x-3)(x+3).
Which is greater: 3/4 or 5/7?
One way to compare fractions is to re-express them with a common denominator. 3/4 = 21/28 and 5/7 = 20/28. 21/28 is greater than 20/28, so 3/4 is greater than 5/7. Another way to compare fractions is to convert them both to decimals. 3/4 converts to 0.75, and 5/7 converts to approximately 0.714.
What do (f-g)(x), (f*g)(x), and fʳ(x) symbolize?
Performing operations on functions is the same as performing operations on numbers. Once you understand the terminology, use your skills of adding, subtracting, multiplying, and dividing polynomials. (f-g)(x) = f(x)-g(x) (f*g)(x) = f(x)*g(x) fʳ(x) = [f(x)]ʳ
A box contains 2 white balls, 6 red balls, and 8 green balls. If a ball is chosen at random, what is the probability that the ball is green?
Probability = # of desired outcomes/total # of possible outcomes If 8 of 16 balls are green, the probability of choosing a green ball is 8/16 or 1/2.
What are the basic SI units for length, mass, time, volume, temperature, and electric current?
QUANTITY/NAME/SYMBOL Length/Meter/m Mass/Kilogram/kg Time/Second/s Volume/Liter/L Temperature/Kelvin/K Electric Current/Ampere/A
Which is larger: 2/3 or (2/3)^2?
Raising a fraction between 0 and 1 to a power yields a number smaller than the original. (2/3)^2 = 4/9. Therefore, two-thirds is larger.
What are the x- and y-intercepts for the following functions? (i) y = 2x + 5 (ii) y = (x - 4)/(x^2 + 2x - 8)
Remember, y = 0 at the x-intercept, and x=0 at the y-intercept (i) x-intercept = -5/2 y-intercept = 5 (ii) x-intercept = 4 y-intercept = 1/2
What are the three major ways to evaluate limits?
Substitution, factoring, and multiplication by the conjugate
The average of 10 numbers is 50. What is the sum of the 10 numbers?
Sum = (Average) * (Number of Terms). If the average of 10 numbers if 50, then they add up to 10*50, or 500.
Evaluate the following expressions: 5! = ? 0! = ?
The "!" in factorials mean multiplication of a descending series from the given number to 1. 5! = 5*4*3*2*1 = 120 By definition, 0! = 1
What is the domain for the following functions? (i) y = 3x - 3 (ii) y = (x+2)/(x^2-3x+2) (iii) y = |x + 4| (iv) y = sqrt(x-1)
The domain is any real number that x can have and that results in the function returning a real value. (i) Domain = all real numbers (ii) Domain = all real numbers except 1 and 2 (iii) Domain = all real numbers (iv) Domain = [1, inf)
List all the factors of 18
The factors of integer n are the positive integers that divide n with no remainder. The multiplies of n are the integers that n divides into with no remainder. 3 is a factor of 6, and 18 is a multiple of 6. All the factors of 18 are 1, 2, 3, 6, 9, and 18
Convert 4 square feet into square inches
The key to converting between units with exponents is raising the conversion to the same power as the unit. 4ft^2/1 * (12^2)*(in^2)/(1^2)*(ft^2) = 576 in^2
Which normal curve has the highest mean and which curve has the highest standard deviation? graph A: 40 bell graph B: 20 tall bell graph C: 30 short bell
The mean is represented at the peak of a normal curve. The highest mean is at the greatest value where the peak lies on the curve, which in this case is graph A. The curve that deviates the most from the mean has the highest standard deviation. That curve is the widest curve, as in graph C.
What is the median of 12, 15, 23, 40, and 40?
The median is the middle value. The median of 12, 15, 23, 40, and 40 is 23 because it's the middle value: two numbers are smaller, and two numbers are bigger. If there's an even number of values, the median is halfway between the two middle values. The median of 12, 15, 23, and 40 is 19 because it's halfway between the two middle values 15 and 23.
What is the mode of 12, 15, 23, 40, and 40?
The mode is the number that appears the most often. The mode of 12, 15, 23, 40, and 40 is 40 because it appears more often than any other number. if two numbers appear equally often, they are both modes.
sqrt3 * sqrt5 = ? sqrt6/sqrt3 = ?
The product of square roots is equal to the square root of the product: sqrt3 * sqrt5 = sqrt(3*5) = sqrt15. The quotient of square roots is equal to the square root of the quotient: sqrt6/sqrt3 = sqrt(6/3) = sqrt2
What is the range for the following functions? (i) y = 3x - 3 (ii) y = x^2 - 4x + 2 (iii) y = |x + 4| (iv) y = sqrt(x-1)
The range is any real result the function can have from the domain. (i) Range = all real numbers (ii) Range = [-2, inf) (iii) Range = [0, inf) (iv) Range = [0, inf)
What is the remainder when 487 is divided by 5?
The remainder is the whole number left over after division. 487 is 2 more than 485, which is a multiple of 5, so when 487 is divided by 5, the remainder will be 2
What is the positive difference between the sum of 4 and 5 and the product of 4 and 5?
The sum is the result of addition. The difference is the result of subtraction. The product is the result of multiplication. The sum of 4 and 5 is 9. The product of 4 and 5 is 20. The positive difference between 9 and 20 is 11.
What do you call the sum or difference of vectors?
The sum or difference of vectors is denoted as the resultant because adding or subtracting vectors results in a new vector.
If a bag contains 3 white balls, 6 green balls, 8 red balls, and 13 blue balls, what is the probability of drawing a white ball, replacing it, and drawing a blue ball?
The word "and" is key to the question. It indicates that we will multiply the probability of drawing a white ball by the probability of drawing a blue ball. Pwhite * Pblue = 3/30 * 13/30 = 39/900 = 13/300
If a bag contains 3 white balls, 6 green balls, 8 red balls, and 13 blue balls, what is the probability of drawing a green or red ball?
The word "or" is key to the question. It indicates that we will add the probability of drawing a green ball to the probability of drawing a red ball. Pgreen + Pred = 6/30 + 8/30 = 14/30 = 7/15.
How do you add and subtract vectors?
To add and subtract vectors, convert them to their respective x- and y-components. Vectors given in rectangular notation are already separated into x- and y-components. Vectors given in any other notation must be converted to x- and y-components by the following formulas: Ax = Acos(theta) Ay = Asin(theta) x-components and y-components are then individually combined.
(2/15) + (3/10) = ?
To add or subtract fractions, first find a common denominator, then add or subtract the numerators. (2/15) + (3/10) = (4/30) + (9/30) = (4+9)/30 = 13/30
(3x^2 + 5x + 7) - (x^2 + 12) = ?
To add or subtract polynomials, combine like terms. (3x^2 + 5x + 7) - (x^2 + 12) = (3x^2 - x^2) + 5x + (7 - 12) = 2x^2 + 5x - 5.
2a + 3a = ?
To combine like terms, keep the variable part unchanged while adding or subtracting the coefficients. 2a + 3a = (2+3)a = 5a
Express 5/8 as a decimal.
To convert a fraction to a decimal, divide the bottom into the top. To convert 5/8, divide 8 into 5, yielding 0.625.
Express 7 1/3 as an improper fraction
To convert a mixed number to an improper fraction, multiple the whole number part by the denominator, then add the numerator. The result is the new numerator (over the same denominator). To convert 7 1/3, first multiple 7 by 3, then add 1, to get the new numerator of 22. Put that over the same denominator, 3, to get 22/3.
Convert 36 km/hr into m/s
To convert units, we have to be able to use our knowledge of prefixes. Then it is just a matter of dimensional analysis. 36km/hr * 1000m/km * 1hr/3600s = 10m/s
How do you divide complex numbers?
To divide complex numbers, we must first multiply the denominator and the numerator by the conjugate of the denominator. This is the same as multiplying by 1 and does not change the equation. It does remove the imaginary part from the denominator son that we get a complex answer to the quotient. Remember, the conjugate of (a+bi) is (a-bi)
Solve 6!/4!
To divide factorials, re-write the numerator as a factorial in terms of the denominator so that you can cancel like terms. (6*5*4!)/4! = 6 * 5 = 30
(1/2)/(3/5) = ?
To divide fractions, invert the second one and multiply. (1/2)/(3/5) = (1/2)*(5/3) = (1*5)/(2*3) = 5/6
Find the inverse of the function y = 2x + 3
To do this, simply switch y and x and solve for y x = 2y+3 y = (x-3)/2
What is the value of x^2 + 5x - 6 when x = -2?
To evaluate an algebraic expression, plug in the given values for the unknowns and calculate according to PEMDAS. To find the value of x^2+5x-6 when x = -2, plug in -2 for x: (-2)^2+5(-2)-6=4+(-10)-6 = -12.
Factor: x^2-5x+6
To factor a quadratic expression, think about what binomials you could use FOIL on to get that quadratic expression. To factor x^2-5x+6, think about what First terms will produce x^2, what Last terms will produce +6, and what Outer and Inner terms will produce -5x. Some common sense and a little trial and error lead you to (x-2)(x-3).
The average of 4 number is 7. If 3 of the numbers are 3, 5, and 8, what is the fourth number?
To find a missing number when you're given the average, use the sum. If the average of 4 numbers is 7, then the sum of those 4 numbers is 4*7, or 28. Three of the numbers (3,5 and 8) add up to 16 of that 28, which leaves 12 for the fourth number.
A basket contains 12 apple and 20 oranges. What is ratio of oranges to apples?
To find a ratio, put the number associated with the word "of" on top and the quantity associated with the word "to" on the bottom and reduce. The ratio of 20 oranges to 12 apples is 20/12, which reduces to 5/3.
What is the average (arithmetic mean) of 12, 15, 23, 40, and 40?
To find the average of a set of numbers, add them up and divide by the number of terms. Avergage = (sum of the terms)/(number of terms) To find the average of the 5 numbers 12, 15, 23, 40, and 40, first add them up: 12 + 15 + 23 + 40 + 40 = 130. Then divide the sum by 5: 130/5 = 26.
What is the average of all the integers from 13 through 77, inclusive?
To find the average of evenyl spaced numbers, just average the smallest and the largest. The average of all the integers from 13 through 77 is the same as the average of 13 and 77. (13+77)/2 = 90/2 = 45
What is the greatest common factor of 36 and 48?
To find the greatest common factor, break down both numbers into their prime factorizations and find the product of all the prime factors they have in common. 36 = 2*2*3*3, and 48 = 2*2*2*2*3. What they have in common is two 2s and one 3, so the GCF is 2*2*3 = 12.
What is the least common multiple of 12 and 15?
To find the least common multiple, check out the multiples of the larger number until you find one that's also a multiple of the smaller. Taking the multiples of 15: 15 is not divisible by 12; 30's not; nor is 45. But the next multiple of 15, 60, is divisible by 12, so it's the LCM.
What is the reciprocal of 3/7?
To find the reciprocal of a faction, switch the numerator and the denominator. The reciprocal of 3/7 is 7/3.
What number is 25% more than 40?
To increase a number by a percent, add the percent to 100%, convert to a decimal, and multiply. To increase 40 by 25%, add 25% to 100%, convert 12% to 1.25, and multiply by 40. 1.25 * 40 = 50
(x + 3)(x + 4) = ?
To multiply binomials, use FOIL. To multiply (x+3) by (x+4), first multiply the First terms: x*x = x^2. Next the Outer terms: x*4 = 4x. Then the Inner terms: 3*x=3x. And finally the Last terms: 3*4=12. Then add and combine like terms: x^2+4x+3x+12=x^2+7x+12.
What is the product of the following complex numbers? z = 3 + 2i w = 1 - 3i
To multiply complex numbers, we use the same method we use to multiply polynomials, in this case FOIL. z * w = (3 + 2i) * (1 - 3i) =3 - 9i + 2i - 6i^2 = 3 - 7i + 6 =9 - 7i
2a * 3a = ?
To multiply monomials, multiply the coefficients and the variables separately. 2a * 3a = (2*3)(a*a) = 6a^2
x^3 * x^4 = ?
To multiply powers, add the exponents: x^3*x^4 = x^(3+4) = x^7 To divide powers, subtract the exponents: y^13/y^8 = y^(13-8) = y^5
(x^3)^4 = ?
To raise a power to an exponent, multiply the exponents. (x^3)^4 = x^(3*4) = x^12
If x^2 + 12 = 7x, what are the possible values of x?
To solve a quadratic equation, put it in the "unfoiled factors = 0" form, factor the left side, and set each factor equal to 0 seperately to get the two solutions. To solve x^2 + 12 = 7x, first re-write it as x^2 - 7x + 12 = 0. Then factor the left side: (x-4)(x-3) = 0 x-4 = 0 or x - 3 = 0 x = 4 or 3
If 3x - 10y = -5x + 6y, what is the value of x in terms of y?
To solve an equation for one variable in terms of another means to isolate the one variable on one side of the equation, leaving an expression containing the other variable on the other side of the equation. To solve the equation 3x - 10y = -5x + 6y for x in terms of y, isolate x: 3x-10y = -5x+6y 3x+5x = 6y+10y 8x = 16y x = 2y
If 5x - 12 = -2x + 9, what is the value of x?
To solve an equation, do whatever is necessary to both sides to isolate the variable. To solve the equation 5x-12 = -2x + 9, first get all the xs on one side by adding 2x to both sides: 7x - 12 = 9. Then add 12 to both sides: 7x = 21. Then divide both sides by 7 to get x = 3.
What is the complete range of values of x for which -5x+7<-3 and -2x+7>1?
To solve an inequality, do whatever is necessary to both sides to isolate the variable. Just remember that when you multiply or divide both sides by a negative number, you must reverse the sign. To solve -5x+7<-3, subtract 7 from both sides to get -5x<-10. Now divide both sides by -5, remembering to reverse the sign: x>2. To solve -2x+7>1, subtract 7 from both sides to get -2x>-6. Now divide both sides by -2 (reversing the sign) to get x<3. Therefore, 2<x<3.
sqrt(4/9) = ?
To take the square root of a fraction, take the square roots of the numerator and denominator separately. sqrt(4/9) = sqrt4/sqrt9 = 2/3
What are the different ways to multiply vectors, and how are they different?
Vectors can be multiplied using the dot product and cross product The dot product returns a scalar value only, whereas the cross product returns a vector value that is perpendicular to the plane of the multiplied vectors (right-hand rule)
How do you denote a vector?
Vectors can be represented in several ways: 1. With an origin and a point: OP 2. As just a variable: A 3. In rectangular notation: 4i + 3j 4. In polar notation: 5, 45deg
What are the derived SI units for velocity, acceleration, density, force, pressure, energy, and power?
Velocity/Meters per Second/(m/s) Acceleration/Meter per Second Squared/(m/s^2) Density/Kilogram per Cubic Meter/(kg/m^3) Force/Newton/N=kg*m*(s^-2) Pressure/Pascal/Pa=N/m^2=kg*m^-1*s^-2 Energy/Joule/J=N*m=kg*m^2*s^-2 Power/Watt/W=J/s=kg*m^2*s^-3
Can horizontal or vertical asymptotes be crossed when graphing a function?
Vertical asymptotes cannot be crossed by a continuous function; horizontal asymptotes can be, and often are, crossed.
Find the vertical asymptotes of the following functions: (i) y = (x+3)/(x^2+2x-15) (ii) y = (2x^2-8)/(x^2+x-12)
Vertical asymptotes occur whenever the function becomes undefined. By definition, vertical asymptote values are not included in the domain. Set the denominators of both functions equal to zero. (i) The denominator factors to (x-5)(x+3). Therefore, x=5 and x= -3 are vertical asymptotes. (ii) The denominator factors to (x+4)(x-3). Therefore, x=-4 and x=3 are vertical asymptotes.
Express 3/10 as a percent?
When converting, remember that 1/10 = 0.1 = 10%. 3/10 is 3 times that, so it equals 30%.
Express 2/3 as a percent.
When converting, remember that 1/3 = 0.333.... = 33 1/3%. 2/3 is twice that, so it equals 66 2/3%
Express 75% as a fraction in lowest terms.
When converting, remember that 1/4 = 0.25 = 25%. 75% is 3 times that, so it equals 3/4.
Express 2/5 as a decimal
When converting, remember that 1/5 = 0.2 = 20%. 2/5 is twice that much, so it equals 0.4.
Express 37 1/2% as a fraction in lowest terms.
When converting, remember that 1/8 = 0.125 = 12 1/2%. 37 1/2% is 3 times that, so it equals 3/8.
9-2*(5-3)^2+6/3 = ?
When performing multiple operations, follow PEMDAS, which means Parentheses first, then Exponents, then Multiplication/Division and last, Addition/Subtraction. In the expression 9-2*(5-3)^2+6/3, begin with the parentheses: (5-3) = 2. Then do exponent: 2^2 = 4. Now the expression is 9-2*4+6/3. Next do the multiplication and division to get 9-8+2, which equals 3
What is the formula for the dot product?
X • Y = |X||Y|cos(theta) where X and Y and are two vectors and theta is the angle between them
If 4x + 3y = 8 and x + y = 3, what is the value of x?
You can solve for two variables only if you have two distinct equations. Combine the equations in such a way that one of the variables cancels out. To solve the two equations 4x + 3y = 8 and x+y=3, multiply both sides of the second equation by -3 to get -3x-3y=-9. Now add the two equations; the 3y and the -3y cancel out, leaving x = -1.
Define the following terms: zeroes of a function, domain, and range
Zeroes of a function: x-intercept, y-intercept, vertical asymptotes, and horizontal asymptotes. Domain: any real number values that x can have to make the function return a real number Range: all possible y-values generated from the domain
(5/7)*(3/4) = ?
(15/28)
