Question

Utilize the Fundamental Theorem of Calculus to:

a) Compute definite integrals
b) Solve optimization problems
c) Evaluate derivatives of integrals
d) Simplify complex functions

Answer 1

The Fundamental Theorem of Calculus allows us to compute definite integrals, solve optimization problems, evaluate derivatives of integrals, and simplify complex functions.

Step-by-step explanation:

Utilizing the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a powerful tool in calculus that allows us to connect integration and differentiation. It has various applications, including:

1. Computing definite integrals: The Fundamental Theorem of Calculus states that if f(x) is continuous on the interval [a, b] and F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a).
2. Solving optimization problems: By utilizing the Fundamental Theorem of Calculus, we can find the critical points of a function and determine whether they correspond to maximum or minimum values.
3. Evaluating derivatives of integrals: The Fundamental Theorem of Calculus also allows us to evaluate the derivative of an integral. This is useful in situations where we need to find the rate of change of a quantity represented by an integral.
4. Simplifying complex functions: The Fundamental Theorem of Calculus can be used to simplify complex functions by expressing them as integrals and then evaluating the integrals.

The Fundamental Theorem of Calculus is crucial in many areas of calculus and provides a fundamental connection between differentiation and integration.