A farmer wants to build a rectangular enclosure. He has 400 feet of fencing. What dimensions will maximize the area of the enclosure?

The position of a particle moving along a line is given by s(t) = t^3 - 6t^2 + 9t. At what time(s) is the particle at rest?

Water is being poured into a conical tank at a rate of 2 cubic meters per minute. The tank is 4 meters tall and has a radius of 2 meters at the top. How fast is the water level rising when the water is 3 meters deep?

Find the absolute minimum value of the function f(x) = x^3 - 3x^2 on the interval [-1, 3].

A spherical balloon is being inflated. If the radius is increasing at a rate of 2 cm/s, how fast is the volume increasing when the radius is 5 cm? (V = (4/3)πr^3)

What does the second derivative test tell us about a critical point where f''(x) > 0?

Find the critical points of the function f(x) = x^4 - 4x^3 + 10.

A rectangle is inscribed in a circle of radius 5. What is the maximum possible area of the rectangle?

A car's position is given by the function x(t) = 2t^3 - 15t^2 + 24t + 10. What is the car's velocity at t = 2?

What is the first derivative used to find?