Conjugate method can only be used when either the numerator or denominator contains exactly two terms. In order to use it, we have to multiply by the conjugate of whichever part of the fraction contains the radical. The conjugate of two terms is those same two terms with the opposite sign in between them. Notice that we multiply both the numerator and denominator by the conjugate, because that’s like multiplying by 1, which doesn’t change the value of the original function.
Read MoreIn this section we’ll talk about how to determine whether a graph represents a one-to-one function. If a relation is a function, then it has exactly one y-value for each x-value. If a function is one-to-one, it also has exactly one x-value for each y-value. The reason we care about one-to-one functions is because only a one-to-one function has an inverse. If the function is not one-to-one, then some restrictions might be needed on the domain of the function to make it invertible. The first way we’ll look at whether or not a function is one-to-one is using the Horizontal Line Test.
Read MoreFunctional notation is a way of writing a function. Traditionally f(x) is how a function is written, but really any variables may be used. The function name is the f or the variable outside of parentheses and the variable used for input is x or the variable that’s inside the parentheses.
Read MoreThink about signed numbers just as positive and negative numbers. Positive numbers have positive signs (even though we often write positive numbers without actually putting a positive sign in front of them), whereas negative numbers have negative signs. So 3, 7, and 11 are all positive numbers, but -2, -6, and -9 are all negative numbers.
Read MoreIn this lesson we’ll look at how to write a converse statement from a conditional. We learned in the last section that a conditional statement is an if/then statement where the first part is the hypothesis and the second part is the conclusion.
Read MoreIndependence of path is a property of conservative vector fields. If a conservative vector field contains the entire curve C, then the line integral over the curve C will be independent of path, because every line integral in a conservative vector field is independent of path, since all conservative vector fields are path independent.
Read MoreWhen the terms of a series decrease toward 0, we say that the series is converging. Otherwise, the series is diverging. The nth term test is inspired by this idea, and we can use it to show that a series is diverging. Ironically, even though the nth term test is one of the convergence tests that we learn when we study sequences and series, it can only test for divergence, it can never confirm convergence.
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 MoreIn this lesson we’ll look at the definition of an inverse function and how to find a function’s inverse. If you remember from the last lesson, a function is invertible (has an inverse) if it’s one-to-one. Now let’s look a little more into how to find an inverse and what an inverse does.
Read MoreThe Cartesian coordinate system is the structure we use to graph points in two dimensions. Something that has two dimensions is a surface. The Cartesian coordinate system (also called the Cartesian plane, or just “the plane”) is a flat surface (like the cover of a book) that extends forever in all directions.
Read MoreThe ratio test for convergence lets us determine the convergence or divergence of a series a_n using a limit, L. Once we find a value for L, the ratio test tells us that the series converges absolutely if L<1, and diverges if L>1 or if L is infinite. The test is inconclusive if L=1. The ratio test is used most often when our series includes a factorial or something raised to the nth power.
Read MoreIf we want to find the acute angle between two lines, we can convert the lines to standard vector form and then use the formula cos(theta)=(a•b)/(|a||b|), where a and b are the given vectors, a•b is the dot product of the vectors, |a| is the magnitude of the vector a (its length) and |b| is the magnitude of the vector b (its length).
Read MoreIn this lesson we’ll look at how the rotation of a figure in a coordinate plane determines where it’s located. A rotation is a type of transformation that moves a figure around a central rotation point, called the point of rotation. The point of rotation can be inside or outside of the figure.
Read MoreThink of the domain of a function as all the real numbers you can plug in for x without causing the function to be undefined. The range of a function is then the real numbers that would result for y from plugging in the real numbers in the domain for x. In other words, the domain is all x-values or inputs of a function, and the range is all y-values or outputs of a function.
Read MoreIn this lesson we’ll learn about the qualities that make up parallel and perpendicular lines and how to identify them on a graph or in an equation. Remember, opposites are numbers with different signs, as a variable they can be expressed as m and -m (although this doesn’t mean that m is the positive number and -m is the negative number).
Read MoreWe’ve been looking at physical applications of derivatives, but there are also economics applications. In this lesson, we’ll look at marginal cost, revenue, and profit. But before we jump into these marginal values, let’s look at cost, revenue, and profit in general.
Read MoreIn this lesson we’ll look at how to use triangle congruence theorems to prove that triangles, or parts of triangles, are congruent to one another. A pair of congruent triangles have exactly the same size and shape. That means that all pairs of sides are the same length and all pairs of angles have the same measure. You know that triangles are congruent to one another if the pairs of sides and angles are congruent. The good news is you don’t have to show that all six pairs match up.
Read MoreIn a composite function, one function is used as a variable in the other function. So to find a composite, we’ll plug one function into the other function, and then simplify the result.
Read MoreWhen you multiply two numbers in scientific notation, you want to follow the same set of steps each time: 1) Multiply their decimal numbers, 2) Multiply their powers of 10. By the rules of exponents, we add the exponents when we do this, and 3) Express the results together in proper scientific notation.
Read MoreIn this lesson we’ll look at exterior angles of polygons and the relationship between those and their corresponding interior angles. An exterior angle of a polygon is an angle that’s supplementary to one of the interior angles of the polygon, has its vertex at the vertex of that interior angle, and is formed by extending one of the two sides of the polygon (at that vertex) in the direction opposite (180º away from) that side.
Read More
