particle moves on a vertical line so that its coordinate at time t is given by 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
505. Find the point on the line
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
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. Two light sources, one five times as strong as the other, are placed 20 feet apart, and an object is placed on the line between them. How far from the bright light source should the object be placed so that the object receives the least illumination?
512. Find the area of the largest rectangle that can be inscribed in the ellipse
Doing Approximations Using Newton’s Method
513–515 Find the fifth approximation of the root of the equation using the given first approximation.
513.
514.
515.
Approximating Roots Using Newton’s Method
516–518 Find the root using Newton’s method.
516. Use Newton’s method to find the root of cos
517. Use Newton’s method to find the root of
518. Use Newton’s method to find the positive root of
Chapter 9
Areas and Riemann Sums
This chapter provides some of the groundwork and motivation for antiderivatives. Finding the area underneath a curve has real-world applications; however, for many curves, finding the area is difficult if not impossible to do using simple geometry. Here, you approximate the area under a curve by using rectangles and then turn to Riemann sums. The problems involving Riemann sums can be quite long and involved, especially because shortcuts to finding the solution do exist; however, the approach used in Riemann sums is the same approach you use when tackling definite integrals. It’s worth understanding the idea behind Riemann sums so you can apply that approach to other problems!
The Problems You’ll Work On
This chapter presents the following types of problems:
Using left endpoints, right endpoints, and midpoints to estimate the area underneath a curve
Finding an expression for the definite integral using Riemann sums
Expressing a given Riemann sum as a definite integral
Evaluating definite integrals using Riemann sums
What to Watch Out For
Here are some things to keep in mind as you do
