The slope of a line is how steep the graph of the line is, or the rate of change of the y-coordinates of the points of the graph as you go from left to right. In the equation of a line, slope is denoted by m. It’s not known why m is used, but you can think “m is for move” to help you remember it’s how fast the graph “moves” (changes).
Read MoreDividing one number by another in scientific notation is really similar to multiplying two numbers in scientific notation, because we’re basically following the same steps. First, we’ll divide their decimal numbers, then we’ll divide their powers of 10. By the rules of exponents, we subtract the exponents when we do this. Finally, we express the results together in proper scientific notation.
Read MoreIn this lesson you’ll learn to solve equations with decimals by multiplying by powers of 10. We can solve an equation with decimals by multiplying the equation by a power of 10, and that’s the first thing we’ll do in order to get rid of the decimals by changing them to whole numbers.
Read MoreIn this lesson we’ll look at how to use conditionals to write a logic chain of statements. We already know that a conditional statement is an if/then statement where the first part is the hypothesis and the second part is the conclusion, like this: “If A, then B.”
Read MoreGreen’s theorem gives us a way to change a line integral into a double integral. If a line integral is particularly difficult to evaluate, then using Green’s theorem to change it to a double integral might be a good way to approach the problem.
Read MoreSometimes we’ll want to use polynomial long division to simplify a fraction, but either the numerator and/or denominator isn’t a polynomial. In this case, we may be able to replace the non-polynomial with its power series expansion, which will be a polynomial. The simplest way to do this for the non-polynomial is to find a similar, known power series expansion and then modify it to match the non-polynomial function.
Read MoreIn this lesson we want to talk about the dimensionality of a vector set, which we should start by saying is totally different than the dimensions of a matrix. For now let’s just say that the dimension of a vector space is given by the number of basis vectors required to span that space.
Read MoreWe can do more than just calculate the probability of pulling exactly 3 red marbles in 5 total pulls. For any binomial random variable, we can also calculate something like the probability of pulling at least 3 red marbles, or the probability of pulling no more than 3 marbles.
Read MoreIn this lesson we’ll look at how to find the area of a parallelogram. A parallelogram is a quadrilateral with two pairs of opposite parallel sides. The area of a parallelogram is found by multiplying the base by the height, so A=bh. The height of a parallelogram needs to be drawn in and is perpendicular to its base.
Read MoreThe area of a trapezoid is given by A=(1/2)(b_1+b_2)h, where b_1 and b_2 are the lengths of the parallel bases, and h is the height of the trapezoid (which is perpendicular to the parallel bases). Sometimes you’ll need to draw in the height once you determine which sides are the parallel bases.
Read MoreThe purpose of number word problems is to give you practice in translating back and forth from words to numbers and vice versa. In a number word problem you’re given information about a pair or group of numbers and you usually need to translate the information into equations to solve for the numbers.
Read MoreTo determine whether two lines are parallel, intersecting, skew, or perpendicular, we’ll test first to see if the lines are parallel. If they aren’t parallel, then we test to see whether they’re intersecting. If they’re intersecting, then we test to see whether they are perpendicular, specifically. If they’re not intersecting, we skip the test for perpendicularity, and then we test for skewness.
Read MoreYou have two options for writing the equation of a line: point-slope form and slope-intercept form. Both of them require that you know at least two of the following pieces of information about the line: 1) a point, 2) another point, 3) the slope, m, or 4) the y-intercept, b (the y-coordinate of the point at which the graph of the line crosses the y-axis).
Read MoreTo calculate the work done when we lift a weight or mass vertically some distance, we’ll use the integration formula for work, where W is the work done, F(x) is the force equation, and [a,b] is the starting and ending height of the weight or mass. Oftentimes problems like these will have us use a rope or cable to lift an object up some vertical height. In a problem like this, we’ll need to determine the combined force required to lift the rope and the object.
Read MoreAs a reminder, signed numbers are positive and negative numbers. When we divide signed numbers, therefore, there are three possible combinations. We could be dividing 1) two positive numbers, 2) two negative numbers, or 3) one positive number and one negative number. When the signs are the same, the quotient will be positive, but when the signs are different, the quotient will be negative.
Read MoreUpper triangular matrices are matrices in which all entries below the main diagonal are 0. The main diagonal is the set of entries that run from the upper left-hand corner of the matrix down to the lower right-hand corner of the matrix. Lower triangular matrices are matrices in which all entries above the main diagonal are 0.
Read MoreBayes’ theorem, also known as Bayes’ law or Bayes’ rule, tells us the probability of an event, given prior knowledge of related events that occurred earlier. To simplify Bayes’ theorem problems, it can be really helpful to create a tree diagram. If you’re ever having trouble figuring out a conditional probability problem, a tree diagram is a great tool to fall back on, because it shows all of the sample space of the problem.
Read MoreIterated integrals are double or triple integrals whose limits of integration are already specified. In this lesson, we’ll look at how to evaluate triple iterated integrals by working from the inside toward the outside.
Read MoreThe optimization process is all about finding a function’s least and greatest values. If we use a calculator to sketch the graph of a function, we can usually spot the least and greatest values. The first part of the optimization investigation is about solving for critical points and then classifying them as representing local or global maxima or minima.
Read MoreThe general term of a sequence an is a term that can represent every other term in the sequence. It relates each term in the sequence to its place in the sequence. To find the general term, a_n, we need to relate the pattern in the sequence of terms to the corresponding value of n.
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