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 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 Fundamental Theorem. by using the 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 antiderivative. using an Finding Area Using Antiderivatives • Find the area of the region between the curve y = 4 – x², 0≤ x ≤ 3, and the x-axis. Homework!!!!! • Textbook – p. 302 – 303 # 1 – 26, 41 – 44.