Speed:
This calculus math tutorial from NutshellMath offers homework help in working with the fundamental theorems of calculus and the mean-value theorem. The fundamental theorem of calculus can be used to evaluate definite integrals, and finding the area underneath a curve. The fundamental theorem states that the integral from points a to b of an equation f(x), with respect to x, is equal to F(b)-F(a), where F(x) is an antiderivative of f(x). This theorem holds when f(x) is continuous on the interval [a,b]. While rather simple, this theorem is very powerful when used to evaluate integrals.
A related theorem is the second fundamental theorem of calculus. This theorem explores the derivative of a definite integral. The second fundamental theorem states that the derivative with respect to x of a definite integral of an equation f(t), integrated with respect to t and integrated from point a to x, is equal to the function value f(x). This theorem is extremely helpful when treating integrals as functions, and can be applied to solve a variety of problems.
The mean value theorem is the third theorem discussed in the tutorial. The mean-value theorem simply states that for an integral from points a to b, of an equation continuous on that interval, will equal the product of the width of interval and a function value of at least one point on that interval. In essence this theorem states that the area under the curve of any continuous function is bounded by the area of the two rectangles created by the width of the interval and the maximum and minimum points of the curve on that interval, and that somewhere on the interval, a function value exists where the product of that value and the width of the interval equals the area under the curve. This theorem offers an idea of an “average value” of a function on a given interval. This concept can be applied to a variety of problems.
This tutorial introduces three key theorems in calculus, all of which are extremely important in solving problems. Mastering these theorems is essential to understanding and mastering calculus.
This calculus math tutorial from NutshellMath offers homework help in working with the fundamental theorems of calculus and the mean-value theorem. The fundamental theorem of calculus can be used to evaluate definite integrals, and finding the area underneath a curve. The fundamental theorem states that the integral from points a to b of an equation f(x), with respect to x, is equal to F(b)-F(a), where F(x) is an antiderivative of f(x). This theorem holds when f(x) is continuous on the interval [a,b]. While rather simple, this theorem is very powerful when used to evaluate integrals.
A related theorem is the second fundamental theorem of calculus. This theorem explores the derivative of a definite integral. The second fundamental theorem states that the derivative with respect to x of a definite integral of an equation f(t), integrated with respect to t and integrated from point a to x, is equal to the function value f(x). This theorem is extremely helpful when treating integrals as functions, and can be applied to solve a variety of problems.
The mean value theorem is the third theorem discussed in the tutorial. The mean-value theorem simply states that for an integral from points a to b, of an equation continuous on that interval, will equal the product of the width of interval and a function value of at least one point on that interval. In essence this theorem states that the area under the curve of any continuous function is bounded by the area of the two rectangles created by the width of the interval and the maximum and minimum points of the curve on that interval, and that somewhere on the interval, a function value exists where the product of that value and the width of the interval equals the area under the curve. This theorem offers an idea of an “average value” of a function on a given interval. This concept can be applied to a variety of problems.
This tutorial introduces three key theorems in calculus, all of which are extremely important in solving problems. Mastering these theorems is essential to understanding and mastering calculus.