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kidia
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If f is continuous over an interval containing (a,x)find from first principles the derivative of the function f(x)=integral f(t)dt.Any help?
The derivative of the integral of a continuous function f(x) is the original function f(x). This is known as the fundamental theorem of calculus. It states that the derivative of the integral of a function is equal to the original function.
This is because the derivative of a function measures the rate of change of that function at a given point. The integral, on the other hand, measures the area under the curve of the function. These two operations are inverses of each other, meaning they undo each other's effects. Therefore, the derivative of the integral is the original function.
Yes, this property holds for all continuous functions. As long as the function is continuous and has a defined derivative, the fundamental theorem of calculus applies.
Yes, the fundamental theorem of calculus has many important applications in mathematics and science. It is used in physics to calculate displacement, velocity, and acceleration, and in economics to calculate marginal cost, revenue, and profit. It also plays a crucial role in the field of differential equations.
The fundamental theorem of calculus connects the concept of area under a curve to the concept of slope. The integral of a function represents the area under the curve, and the derivative of the integral gives the slope of the curve. This relationship allows us to use integration to solve problems involving area and slope, making it a powerful tool in many fields.