A Pythagorean triple is a set of three positive integers, (a, b, c), such that a right triangle can be formed with the legs a and b and the hypotenuse c. The most common Pythagorean triples are (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25).
A:The definition of a limit in calculus is the value that a function gets close to but never surpasses as the input changes. Limits are one of the most important aspects of calculus, and they are used to determine continuity and the values of functions in a graphical sense.
A:The abbreviations "sin," "cos," "tan," "csc," "sec" and "cot" stand for the six trigonometric functions: sine, cosine, tangent, cosecant, secant and cotangent. Each function represents a particular relationship between the measure of one of the angles and the ratio between two sides of a right triangle.
A:To calculate bulk density, simply weigh the sample and divide its mass by its volume. Bulk density is commonly used when referring to solid mixtures like soil. Just like particle density, bulk density is also measured in mass per volume.
A:A Riemann sum is a method of approximating the area under the curve of a function. It adds together a series of values taken at different points of that function and multiplies them by the intervals between points. The midpoint Riemann sum uses the x-value in the middle of each of the intervals.
A:The class midpoint, or class mark, is calculated by adding the lower and upper limits of the class and dividing by two. The class midpoint is sometimes used as a representation of the entire class.
A:One thousand millimeters is equal to 1 meter. The meter is the standard unit of length in the International System of Units, also known as the metric system. "Metre" is the standard spelling for all English-speaking countries except the United States. "Meter" is the accepted U.S. spelling.
A:A Pythagorean triple is a set of three positive integers, (a, b, c), such that a right triangle can be formed with the legs a and b and the hypotenuse c. The most common Pythagorean triples are (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25).
A:A nonlinear function in math creates a graph that is not a straight line, according to Columbia University. Three nonlinear functions commonly used in business applications include exponential functions, parabolic functions and demand functions. Quadratic functions are common nonlinear equations that form parabolas on a two-dimensional graph.
A:According to class notes from Bunker Hill Community College, calculus is often used in medicine in the field of pharmacology to determine the best dosage of a drug that is administered and its rate of dissolving. Usually, the drug is slowly dissolved in the stomach.
A:A few examples of how logarithms are used in the real world include measuring the magnitude of earthquakes or the intensity of sound and determining acidity. A logarithm explains how many times a number is multiplied to a power to reach another number. It is expressed as loge(x) and is commonly written as ln(x).
A:The derivative of y = xln(x) with respect to x is dy/dx = ln(x) + 1. This result can be obtained by using the product rule and the well-known results d(ln(x))/dx = 1/x and dx/dx = 1.
A:Single variable calculus covers derivative and integral functions that contain one variable. Functions that contain two or more variables are a part of multivariable calculus.
A:The integral of cosine squared is equal to x/2 + (1/4)sin(2x) + C, where C is a constant. The integral can be solved by using the half-angle identity of the cosine squared function, where cos2(x) = 1/2 + (1/2)cos(2x).
A:The derivative of e^(3x) is equal to three times e to the power of three x. In mathematical terms, the equation can be expressed as d/dx e^(3x) = 3e^(3x).
A:In calculus, critical points or stationary points are any values of differentiable functions of complex or real variables whose derivative is 0, f(x0) = 0. In a differentiable function that has several real variables, critical points are values in the domain where the partial derivatives are 0. The values of critical points are known as critical values.
A:The coefficient of area expansion is a factor that relates change in a material’s external area to temperature. Materials generally expand as they are heated, causing their overall surface area to increase. The extent of this area increase for each degree on a temperature scale is the area expansion coefficient.
A:Calculus was developed independently by both Isaac Newton and Gottfried Leibniz during the later part of the 1600s. For Newton, calculus was primarily a tool he needed for explaining the motion of the planets. It would be difficult to say precisely how he developed his ideas because he was secretive about his methods, but it certainly grew out of his understanding of the laws of motion and acceleration.
A:There are several programs available to help with calculus, though many of them are paid-for products, and none guarantees quick results. However, free online resources for dedicated students include help from Calculus.org and free online courses from MIT. These include multivariable calculus, calculus with theory and calculus with application.
A:The derivative of the expression ln(x) is equal to 1/x. This can be demonstrated by manipulating the equation y = ln(x) into the form x = e^y and taking the derivative of both sides with respect to x.
A:Printable worksheets, online games and online practice exercises are some effective ways for second graders to practice regrouping. Hands-on activities, such as regrouping with counters or base-ten blocks, pretend shopping and recording sports scores also provide students with engaging opportunities to sharpen their regrouping skills.
A:The formula for the integral of inverse tangent is the integral of arctan(x) dx = x * arctan(x) - (1/2) * ln |x2+1|+ C. The integral is solved using integration by parts, which notes that the integral of u dv is equal to u times v minus the integral of v du. The term arctan represents the inverse function in mathematical formulas.