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

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Applying the Mean Value Theorem to Solve Problems

       487– 489 Solve 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 .

Relating Velocity and Position

       490–492 Use 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

Finding Velocity and Speed

       493– 497 Solve 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.

Solving Optimization Problems

       498–512 Solve 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.

      504. Find the point(s) on the ellipse farthest from .

505. Find the point on the line that is closest to the origin.

      506. A rectangular poster is to have an area of 90 square inches with 1-inch margins at the bottom and sides and a 3-inch margin at the top. What dimensions give you the largest printed area?

      507. At which x values on the curve does the tangent line have the largest slope?

      508. A rectangular storage container with an open top is to have a volume of 20 cubic meters. The length of the base is twice the width. The material for the base costs $20 per square meter. The material for the sides costs $12 per square meter. Find the cost of the materials for the cheapest such container. Round your answer to the nearest cent.

      509. A piece of wire that is 20 meters long is cut into two pieces. One is shaped into a square, and the other is shaped into an equilateral triangle. How much wire should you use for the square so that the total area is at a maximum?

      510. A piece of wire that is 20 meters long is cut into two pieces. One is bent into a square, and the other is bent into an equilateral triangle. How much wire should you use for the square so that the total area is at a minimum?

      511. The illumination of a light source is directly proportional to the strength of the light source and inversely proportional to the square of the distance from the source.