Chapter 2
Derivatives and Integrals
By Boundless
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The use of differentiation makes it possible to solve the tangent line problem by finding the slope
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Differentiation is a way to calculate the rate of change of one variable with respect to another.
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If every point of a function has a derivative, there is a derivative function sending the point
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The rules of differentiation can simplify derivatives by eliminating the need for complicated limit calculations.
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Derivatives of trigonometric functions can be found using the standard derivative formula.
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The chain rule is a formula for computing the derivative of the composition of two or more functions.
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Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.
Differentiation, in essence calculating the rate of change, is important in all quantitative sciences.
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Related rates problems involve finding a rate by relating that quantity to other quantities whose rates of change are known.
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The derivative of an already-differentiated expression is called a higher-order derivative.
A linear approximation is an approximation of a general function using a linear function.
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Maxima and minima are critical points on graphs and can be found by the first derivative and the second derivative.
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The MVT states that for a function continuous on an interval, the mean value of the function on the interval is a value of the function.
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The shape of a graph may be found by taking derivatives to tell you the slope and concavity.
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The asymptotes are computed using limits and are classified into horizontal, vertical and oblique depending on the orientation.
Curve sketching is used to produce a rough idea of overall shape of a curve given its equation without computing a detailed plot.
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Graphics can be created by hand, using computer programs, and with graphing calculators.
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Mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives.
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Newton's Method is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
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The second derivative test is a criterion for determining whether a given critical point is a local maximum or a local minimum.
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Differentials are the principal part of the change in a function
An antiderivative is a differentiable function
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Defined integrals are used in many practical situations that require distance, area, and volume calculations.
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A definite integral is the area of the region in the
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The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function to the concept of the integral.
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An indefinite integral is defined as
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Integration by substitution is an important tool for mathematicians used to find integrals and antiderivatives.
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A transcendental function is a function that is not algebraic.
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Numerical integration is a method of approximating the value of a definite integral.
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The area between the graphs of two functions is equal to the integral of a function,
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Volumes of complicated shapes can be calculated using integral calculus if a formula exists for the shape's boundary.
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The average of a function
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In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.
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Forces may do work on a system. Work done by a force (
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Disc and shell methods of integration can be used to find the volume of a solid produced by revolution.