Presentation on theme: "5.4 Fundamental Theorem of Calculus. It is difficult to overestimate the power of the equation: It says that every continuous function f is the derivative."— Presentation transcript:
It is difficult to overestimate the power of the equation: It says that every continuous function f is the derivative of some other function, namely. It says that every continuous function has an antiderivative. It says that the processes of integration and differentiation are inverses of one another.
Applying the Fundamental Theorem Find by using the Fundamental Theorem.
The Fundamental Theorem with the Chain Rule Find dy/dx if
Variable Lower Limits of Integration Find dy/dx.
Constructing a Function with a Given Derivative and Value Find a function y = f(x) with derivative that satisfies the condition f(3) = 5. Since y(3) = 0, we have only to add 5 to this function to construct one with derivative tan x whose value at x = 3 is 5:
The second part of the Fundamental Theorem of Calculus shows how to evaluate definite integrals directly from antiderivatives.
Evaluating an Integral Evaluate using an antiderivative.
Finding Area Using Antiderivatives Find the area of the region between the curve y = 4 – x², 0≤ x ≤ 3, and the x-axis.