Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice). Patrick Jones
Читать онлайн книгу.
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 the problems in this chapter:
Estimating the area under a curve typically involves quite a bit of arithmetic but shouldn’t be too difficult conceptually. The process should be straightforward after you do a few problems.
The problems on expressing a given Riemann sum as a definite integral don’t always have unique solutions.
To evaluate the problems involving Riemann sums, you need to know a few summation formulas. You can find them in any standard calculus text if you don’t remember them — or you can derive them!
Calculating Riemann Sums Using Left Endpoints
519–522 Find the Riemann sum for the given function with the specified number of intervals using left endpoints.
519.
520.
521.
522.
Calculating Riemann Sums Using Right Endpoints
523–526 Find the Riemann sum for the given function with the specified number of intervals using right endpoints.
523.
524.
525.
526.
Calculating