Because exponential and logarithmic functions are inverses of one another, if we have the graph of the exponential function, we can find the corresponding log function simply by reflecting the graph over the line y=x, or by flipping the x- and y-values in all coordinate points. Let’s use some graphs from the previous section to illustrate what we mean.
Read MoreTo find the scalar equation of a line, we’ll use the formulas x=x_0+at, y=y_0+bt, and z=z_0+ct, where P_0(x_0,y_0,z_0) is a given point and v=(a,b,c) is the given vector. The vector may also be in the format v=ai+bj+ck.
Read MoreThe Mean Value Theorem for integrals tells us that, for a continuous function f(x), there’s at least one point c inside the interval [a,b] at which the value of the function will be equal to the average value of the function over that interval. This means we can equate the average value of the function over the interval to the value of the function at the single point.
Read MoreIn this lesson we’ll look at the triangle side splitting theorem and how it relates to solving for missing pieces of information in the triangles. A triangle can be split by a line segment at any spot in the triangle. As long as the segment touches two sides of the triangle, and is parallel to the side it doesn’t touch, then the segment splits the triangle proportionally.
Read MoreWith scientific notation, we’re able to express really really big numbers without all the digits and write a simpler number that quickly tells us how big the number is. So scientific notation can be used to express really, really big numbers, but it can also be used to express really, really small numbers.
Read MoreThe key is that all the terms of the polynomial need to share the factor being taken out. Any factor that’s shared by all the terms is called a common factor, and the factor that consists of everything which is shared by all of them is known as the greatest common factor. Factoring is “un-distributing,” which means that we do the opposite of distributing and take out (or “factor out”) the same factor from each term of the polynomial (and divide each term by that factor to get “what’s left” once it’s taken out).
Read MoreAny vector v that satisfies T(v)=(lambda)(v) is an eigenvector for the transformation T, and lambda is the eigenvalue that’s associated with the eigenvector v. The transformation T is a linear transformation that can also be represented as T(v)=A(v).
Read MorePreviously we talked about a transformation as a mapping, something that maps one vector to another. So if a transformation maps vectors from the subset A to the subset B, such that if ‘a’ is a vector in A, the transformation will map it to a vector ‘b’ in B, then we can write that transformation as T: A—> B, or as T(a)=b.
Read MoreInverse operations are operations that are opposite or “undo” each other. For example, addition undoes subtraction and division undoes multiplication. Inverse operations are useful when solving equations.
Read MoreIn this lesson we’ll look at how to find the midpoint of a line segment in three dimensions when we’re given the endpoints of the line segment as coordinates in three-dimensional space.
Read MoreLine graphs are really similar to bar graphs. In fact, to turn a bar graph into a line graph, all you have to do is connect the middle of the top of each bar to the middle of the top of the bar beside it with a straight line, and you’ll form the line graph. Ogives are like cumulative line graphs.
Read MoreIn this lesson we’ll learn how to calculate a percent of, and the new sale price of an item with a discount. What is a percent markdown? In retail, a store will discount an item in order to sell it more quickly. The store usually does this as a percentage off the original amount they had planned to charge, and if that’s the case we call this a percent markdown.
Read MoreWe say that two vectors a and b are orthogonal if they are perpendicular (their dot product is 0), parallel if they point in exactly the same or opposite directions, and never cross each other, otherwise, they are neither orthogonal or parallel. Since it’s easy to take a dot product, it’s a good idea to get in the habit of testing the vectors to see whether they’re orthogonal, and then if they’re not, testing to see whether they’re parallel.
Read MoreThe goal is to use differentiation to get the left side of this equation to match exactly the function we’ve been given. When we differentiate, we have to remember to differentiate all three parts of the equation. We’ll try to simplify the sum on the right as much as possible, and the result will be the power series representation of our function. If we need to, we can then use the power series representation to find the radius and interval of convergence.
Read MoreThe intermediate value theorem is a theorem we use to prove that a function has a root inside a particular interval. The root of a function, graphically, is a point where the graph of the function crosses the x-axis. Algebraically, the root of a function is the point where the function’s value is equal to 0.
Read MoreThroughout Linear Algebra, we’ll be really interested in solving systems of linear equations, or linear systems. A linear system is a system of equations, defined for a set of unknown variables, where each of the variables is linear (the variables are first degree, or raised to the power of 1).
Read MoreWhen we’re given two vectors with the same initial point, and they’re different lengths and pointing in different directions, we can think about each of them as a force. The longer the vector, the more force it pulls in its direction.
Oftentimes we want to be able to find the net force of the two vectors, which will be a third vector that counterbalances the force and direction of the other two. Think about the resultant vector as representing the amount of force and the direction in which you’d have to pull to cancel out the force from the other two vectors.
Read MorePreviously we learned how to create a power series representation for a function by modifying a similar, known series to match the function. When we have the product of two known power series, we can find their product by multiplying the expanded form of each series in the product.
Read MoreIn this lesson we’ll look at how to find the measures of the interior angles of polygons. We’ll name polygons based on the number of sides, and then talk about the number of triangles that make up the polygon, and how to find the measure of each interior angle.
Read MoreEvaluating expressions means that you’ll be replacing or “plugging in” numbers for variables and then simplifying using the order of operations until you arrive at a single number. Sometimes you’ll be able to plug in the numbers without issue, but there are other times (when there’s multiplication, exponents, or when plugging in a negative number) where you’ll need to plug in the numbers using parentheses.
Read More
