The interval of convergence of a series is the set of values for which the series is converging. Remember, even if we can find an interval of convergence for a series, it doesn’t mean that the entire series is converging, only that the series is converging in the specific interval. The radius of convergence of a series is always half of the interval of convergence. You can remember this if you think about the interval of convergence as the diameter of a circle.
Read MoreJust as we did with single variable functions, we can use the second derivative test with multivariable functions to classify any critical points that the function might have. To use the second derivative test, we’ll need to take partial derivatives of the function with respect to each variable. Once we have the partial derivatives, we’ll set them equal to 0 and use these as a system of simultaneous equations to solve for the coordinates of all possible critical points.
Read MoreYou’ve already learned several ways to classify a whole number: as positive, negative, or 0; as the opposite of a number, in terms of the whole numbers by which it’s divisible, and in terms of its multiples. Another way to classify a whole number greater than 1 is as either a “prime” number or a “composite” number. A whole number will either be a prime number or a composite number, but it can’t be both.
Read MoreA histogram, also called a frequency histogram, is just like a bar graph, except that we collect the data into buckets or bins, and then sketch a bar for each bucket. Each bucket needs to be the same size so that they’re capable of holding the same amount of data. A stem-and-leaf plot (also called a stem plot), is just another way to summarize data. It’s similar to a histogram, because both types of charts group together data points, and are good ways to show how many data points fall into a certain category or range.
Read MoreIn order for a differential equation to be called an exact differential equation, it must be given in the form M(x,y)+N(x,y)(dy/dx)=0. To find the solution to an exact differential equation, we’ll 1) Verify that My=Nx to confirm the differential equation is exact, 2) Use Psi=int M(x,y) dx or Psi=int N(x,y) dy to find Psi(x,y), including a value for h(y) or h(x), and then 3) Set Psi(x,y)=c to get the implicit solution.
Read MoreWe previously learned how to find the gradient vector at a specific point. To find the equation of the tangent plane, we can just use the formula for the gradient vector where (x,y) is the point we’re interested in.
Read MoreThe arc length of a polar curve is simply the length of a section of a polar parametric curve between two points a and b. We use a specific formula in terms of L, the arc length, r, the equation of the polar curve, (dr/dtheta), the derivative of the polar curve, and a and b, the endpoints of the section.
Read MoreWhen we talk about the “divisibility” of a whole number, we’re just talking about the whole numbers that divide evenly into it. As an example, 5 divides evenly into 15 three times, since 15/5=3, which means we can say that 15 is “divisible” by 5. In this lesson we’ll look at divisibility rules for the numbers 2-10.
Read MoreIf we want to find the acute angle between two curves, we’ll find the tangent lines to both curves at their point(s) of intersection, convert the tangent lines to standard vector form and then use the formula for the acute angle, given by cos(theta), 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 MoreThe order of operations is the set of rules that you use to simplify expressions in math. PEMDAS or “Please Excuse My Dear Aunt Sally” are ways to help you remember the order of operations. The order of operations tells us to simplify expressions following this order: Parentheses (all symbols of inclusion), Exponents (powers and roots), Multiplication/Division (from left to right or divide first), then Addition/Subtraction (from left to right or subtract first).
Read MoreWe’re familiar now with displaying data in a two-way table. But sometimes it’s helpful to express the data in a two-way table as percentages. If we want to express percentages, then we just need to change the two-way table into a relative frequency table, which is a table that shows percentages instead of actual counts.
Read MoreIn this lesson you’ll learn how to find the molar mass of chemical compounds given the molar mass of individual atoms, and how to find the molar mass of individual atoms given the molar mass of a compound or molecule.
Read MoreWe already know how to solve an initial value problem for a second-order homogeneous differential equation. Boundary value problems are very similar, but differ in a few important ways: 1) Initial value problems will always have a solution; boundary value problems may not, 2) The initial conditions given in an initial value problem relate to the general solution and its derivative; the initial conditions in a boundary value problem both relate to the general solution, not its derivative, and 3) The initial conditions given in an initial value problem are both for values of x0=0; the initial conditions given in a boundary value problem are for x0=a and x0=b.
Read MoreIn this lesson we’ll look at how to use the slopes of two lines in the Cartesian plane (the xy-plane) to see if the lines are perpendicular, parallel, or neither. Parallel lines have equal slopes and will never intersect each other because they’ll always be the same distance apart. Perpendicular lines have slopes that are negative reciprocals, and they intersect each other at a right angle.
Read MoreUndetermined coefficients is a method you can use to find the general solution to a second-order (or higher-order) nonhomogeneous differential equation. Remember that homogenous differential equations have a 0 on the right side, where nonhomogeneous differential equations have a non-zero function on the right side.
Read MoreLong division of polynomials uses the same steps you learned for long division of real numbers. It might look different because of the variables but don’t worry, it’s the same thing in disguise. Let’s first review long division. Remember this? You followed a pattern of Divide, Multiply, Subtract, Bring Down.
Read MoreIn this lesson we’ll look at inscribed angles of circles and how they’re related to arcs, called intercepted arcs. A chord is a straight line segment that has endpoints on the circumference of the circle, and an inscribed angle is formed by two chords. These chords share the vertex of an angle. The arc that touches the endpoints of the chords is called the intercepted arc.
Read MoreWe can use the formula for the sum of a geometric series to quickly and accurately convert a repeating decimal into a ratio of integers, in other words, into a fraction with whole numbers in the numerator and denominator.
Read MoreHow can you describe the relationship between two numbers? There are only three ways to describe this relationship, which is why it’s called the law of trichotomy (“tri” means three). The law of trichotomy says that two numbers can have exactly one of three possible relationships: 1) a<b, 2) a>b, or 3) a=b.
Read MoreVector, parametric, and symmetric equations are different types of equations that can be used to represent the same line. We use different equations at different times to tell us information about the line, so we need to know how to find all three types of equations.
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