Question

Apply the Fundamental Theorem of Calculus:

a) Finding antiderivatives
b) Evaluating definite integrals
c) Calculating derivatives of definite integrals
d) Solving differential equations

Answer 1

The Fundamental Theorem of Calculus is a powerful tool in calculus that has multiple applications in finding antiderivatives, evaluating definite integrals, calculating derivatives of definite integrals, and solving differential equations. Hence all of the above are correct.

Step-by-step explanation:

The Fundamental Theorem of Calculus is a powerful tool in calculus that has multiple applications. Let's look at each part of the question:

1. Find the antiderivative: The Fundamental Theorem of Calculus allows us to find the antiderivative of a function. This is useful for finding functions whose derivatives are known.
2. Evaluate definite integrals: The Fundamental Theorem of Calculus provides a method for evaluating definite integrals. It relates the value of the definite integral to the antiderivative of the function being integrated.
3. Calculate derivatives of definite integrals: The Fundamental Theorem of Calculus also allows us to calculate the derivative of a definite integral. This is useful for finding rates of change or slopes of curves.
4. Solve differential equations: The Fundamental Theorem of Calculus can be applied to solve differential equations. It relates a function to its derivative and allows us to find solutions to differential equations.

Overall, the Fundamental Theorem of Calculus is a fundamental concept in calculus that has many applications in finding antiderivatives, evaluating definite integrals, calculating derivatives of definite integrals, and solving differential equations.