Calculus: Derivatives & Integrals

What is the derivative of $f(x) = x^3$?

Which integral represents the area under the curve of $f(x) = 2x$ from $x=0$ to $x=3$?

A function $f(x)$ has a derivative $f'(x)$ that changes from positive to negative at $x=5$. What does this indicate about $f(x)$ at $x=5$?

Which technique is most appropriate to find the maximum volume of a box without a top made from a square sheet by cutting out squares from corners?

The limit $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ equals:

If the rate of change of position with respect to time is velocity, what does the derivative of velocity with respect to time represent?

The integral of a function gives the exact area under its curve between two points.

A limit must always exist for a function to be continuous at a point.

The derivative of a constant function is zero.

Explain how derivatives can be used to solve optimization problems in real-world contexts.

Match each calculus concept to its correct application or description: