Calculus

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).

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  • What is the precise definition of a limit in calculus?

    Q: What is the precise definition of a limit in calculus?

    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.
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  • How is calculus used in medicine?

    Q: How is calculus used in medicine?

    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.
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  • How are logarithms used in the world?

    Q: How are logarithms used in the world?

    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).
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  • How do you calculate bulk density?

    Q: How do you calculate bulk density?

    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.
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  • How do you calculate the midpoint Riemann sum?

    Q: How do you calculate the midpoint Riemann sum?

    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.
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  • What are the meanings of "sin", "cos", "tan", "csc", "sec" and "cot"?

    Q: What are the meanings of "sin", "cos", "tan", "csc", "sec" and "cot"?

    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.
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  • What are some common Pythagorean triples?

    Q: What are some common Pythagorean triples?

    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).
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  • What is a nonlinear function in math?

    Q: What is a nonlinear function in math?

    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.
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  • How is the class midpoint calculated?

    Q: How is the class midpoint calculated?

    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.
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  • Q: What is the derivative of e^-x?

    A: The derivative of e-x is -e-x. The derivative of e-x is found by applying the chain rule of derivatives and the knowledge that the derivative of ex is always ex, which can be found using a more complicated proof.
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  • Q: What is integral cos 2?

    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).
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  • Q: How do you explain a gradient vector field in calculus?

    A: Lamar University explains that a gradient vector field is a directional derivative that represents the change in the gradient of a function. Gradient fields are known as conservative fields since they depend only on the beginning point and the end point of the vector while ignoring the path taken by the vector. Another term for gradient vector fields is path-independent vectors.
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  • Q: What is the Lagrange remainder?

    A: The Lagrange remainder is a term that estimates the size of the error introduced by a partial sum of the Taylor series when it is used to compute an approximated value for the original function. The Lagrange remainder for the Taylor series is a tool not only for estimating errors but for giving a precise difference between the polynomial and the original function approximated by the polynomial.
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  • Q: What is the roster method?

    A: The term "roster method" refers to a technique in representing a set by directly listing all of its elements, which are separated by commas and enclosed by a pair of curly brackets. The roster method is also referred to as the "tabular form method."
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  • Q: What are some calculus classes you can take online?

    A: Learners can take calculus courses online through Massachusetts Institute of Technology, San Francisco State University, University of California in Berkeley or Brigham Young University. These courses range from introductory offerings to specific, advanced focuses.
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  • Q: What are some tips on evaluating indefinite integrals?

    A: An indefinite integral is an integral with no limits. Evaluating indefinite integrals can turn out to be very easy if you just stick to a few basics.
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  • What is the prime factorization of 68?

    Q: What is the prime factorization of 68?

    A: The prime factorization of 68 is 2 x 2 x 17. There is only one set of prime factorials for any number. The set of prime factorials is the set of prime numbers that when multiplied together get the original number.
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  • Q: What is the average rate of change in calculus?

    A: The average rate of change in calculus refers to the slope of a secant line that connects two points. In calculus, this equation often involves functions, as opposed to simple points on a graph, as is common in algebraic problems related to the rate of change.
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  • What is single variable calculus?

    Q: What is single variable calculus?

    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.
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  • Q: What are entries as they apply to a matrix?

    A: The items contained in a matrix are its entries; an entry is a single piece of data from within a matrix. Matrix notation refers to the use of a subscript to identify the row and column location of a single entry within a matrix.
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  • What is the spring constant equation?

    Q: What is the spring constant equation?

    A: The equation for the spring force is k = -F/x. This equation is derived from Hooke’s Law, which states that the force needed to extend or compress a spring a certain distance is proportional to that distance. The equation for Hooke’s Law is F=kx, where k is the spring constant.
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