Patrick Jones - Calculus - 1001 Practice Problems For Dummies (+ Free Online Practice)

485. ,

486. ,

487– 489Solve the problem related to the mean value theorem.

487. If and for , what is the smallest possible value of ? Assume that f satisfies the hypothesis of the mean value theorem.

488. Suppose that for all values of x. What are the strictest bounds you can put on the value of ? Assume that f is differentiable for all x.

489. Apply the mean value theorem to the function on the interval to find bounds for the value of .

490–492Use the position function s(t) to find the velocity and acceleration at the given value of t. Recall that velocity is the change in position with respect to time and acceleration is the change in velocity with respect to time.

490. at

491. at

492. at

493– 497Solve the given question related to speed or velocity. Recall that velocity is the change in position with respect to time.

493. A mass on a spring vibrates horizontally with an equation of motion given by , where x is measured in feet and t is measured in seconds. Is the spring stretching or compressing at ? What is the speed of the spring at that time?

494. A stone is thrown straight up with the height given by the function , where s is measured in feet and t is measured in seconds. What is the maximum height of the stone? What is the velocity of the stone when it’s 20 feet above the ground on its way up? And what is its velocity at that height on the way down? Give exact answers.

495. A stone is thrown vertically upward with the height given by , where s is measured in feet and t is measured in seconds. What is the maximum height of the stone? What is the velocity of the stone when it hits the ground?

496. A particle moves on a vertical line so that its coordinate at time t is given by for . When is the particle moving upward, and when is it moving downward? Give an exact answer in interval notation.

497. A particle moves on a vertical line so that its coordinate at time t is given by for . When is the particle moving upward, and when it is it moving downward? Give your answer in interval notation.

498–512Solve the given optimization problem. Recall that a maximum or minimum value occurs where the derivative is equal to zero, where the derivative is undefined, or at an endpoint (if the function is defined on a closed interval). Give an exact answer, unless otherwise stated.

498. Find two numbers whose difference is 50 and whose product is a minimum.

499. Find two positive numbers whose product is 400 and whose sum is a minimum.

500. Find the dimensions of a rectangle that has a perimeter of 60 meters and whose area is as large as possible.

501. Suppose a farmer with 1,500 feet of fencing encloses a rectangular area and divides it into four pens with fencing parallel to one side. What is the largest possible total area of the four pens?

502. A box with an open top is formed from a square piece of cardboard that is 6 feet wide. Find the largest volume of the box that can be made from the cardboard.

503. A box with an open top and a square base must have a volume of 16,000 cubic centimeters. Find the dimensions of the box that minimize the amount of material used.