Which two fundamental concepts of Calculus are inverses of each other? Select the two correct answers.

Question 6Select one or more:

a.

Exponents

b.

Factors

c.

Logarithms

d.

Limits

e.

Derivatives

f.

Integrals

The two fundamental concepts of calculus that are inverses of each other are:

e. Derivatives  
f. Integrals

### Explanation

In calculus, the derivative of a function gives the rate at which the function is changing at any given point, while the integral of a function provides the accumulation of quantities, which can be thought of as the area under the curve of the function. The Fundamental Theorem of Calculus establishes that these two operations are inverses of each other.

**Fundamental Theorem of Calculus**:
1. **First part**: If \( F \) is an antiderivative of \( f \) on an interval \( [a, b] \), then
\[
\int_a^b f(x) \, dx = F(b) - F(a)
\]
2. **Second part**: If \( f \) is continuous on \( [a, b] \), then the function \( F \) defined by
\[
F(x) = \int_a^x f(t) \, dt
\]
for \( x \) in \( [a, b] \), is continuous on \( [a, b] \), differentiable on \( (a, b) \), and \( F'(x) = f(x) \).

Would you like more details or have any questions?

Here are 8 related questions for further practice:
1. What is the relationship between the derivative and the tangent line to a curve?
2. How do you interpret the definite integral geometrically?
3. What is the power rule for differentiation?
4. How do you find the area between two curves using integrals?
5. What is the chain rule in differentiation?
6. How can you apply integration to find the volume of a solid of revolution?
7. What is the product rule in differentiation?
8. How does the concept of a limit underpin the definition of a derivative?

**Tip:** When solving calculus problems, understanding the relationship between derivatives and integrals can help simplify complex problems by converting between rates of change and accumulated quantities.

Explore the fundamental concepts of calculus: derivatives and integrals. Learn how these concepts are inverses of each other according to the Fundamental Theorem of Calculus. Suitable for advanced high school and college-level students.

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