Take the derivative to check your answer. Is it correct?thegoosegirl42 said:I tried taking it to make it -x^-3 - 1x as the antiderivative
Gosh dang it I was going down in exponent when I should have been going up. The new derivative would be -3x^-1 -1x making the answer be -3.5+4 Thus the answer is .5. Thank you so much.Nathanael said:Take the derivative to check your answer. Is it correct?
The Fundamental Theorem of Calculus (FTC) is a fundamental concept in calculus that connects the two main branches of calculus: differential calculus and integral calculus. It states that the definite integral of a function can be calculated by evaluating the difference between the antiderivative of the function at the upper and lower limits of integration.
The FTC is significant because it provides a powerful tool for evaluating integrals and solving a wide range of problems in mathematics, physics, engineering, and other fields. It also allows for the calculation of areas and volumes, as well as the determination of average values of functions.
The FTC is used in various real-world applications, such as computing areas and volumes in engineering and physics, finding probabilities in statistics, and analyzing data in economics and finance. It also has applications in fields like computer science, where it is used in algorithms for solving optimization problems.
The FTC is divided into two parts: Part 1, which relates the derivative and the indefinite integral, and Part 2, which relates the definite integral and the indefinite integral. Part 1 states that if a function is continuous on an interval, then the derivative of the indefinite integral of the function is equal to the original function. Part 2 states that the definite integral of a function can be calculated by evaluating the difference between the antiderivative of the function at the upper and lower limits of integration.
The Fundamental Theorem of Calculus and the Mean Value Theorem are two fundamental theorems in calculus, but they have different purposes. The FTC is used to relate the integral and the derivative of a function, while the Mean Value Theorem is used to show the existence of a point where the slope of a tangent line to a curve is equal to the average slope of the curve on a given interval. In other words, the FTC deals with the area under a curve, while the Mean Value Theorem deals with the slope of a curve at a specific point.