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The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound $ K $ for $ \mid f^{(4)}(x) \mid $ by hand. But computer algebra systems have no problem computing $ f^{(4)} $ and graphing it, so we can easily find a value for $ K $ from a machine graph. This exercise deals with approximations to the integral $ I = \displaystyle \int_0^{2 \pi} f(x)\ dx $,where $ f(x) = e^{\cos x} $.(a) Use a graph to get a good upper bound for $ \mid f^{\prime \prime}(x) \mid $.(b) Use $ M_{10} $ to approximate $ I $.(c) Use part (a) to estimate the error in part (b).(d) Use the built-in numerical integration capability of your $ CAS $ to approximate $ I $.(e) How does the actual error compare with the error estimate in part (c)?(f) Use a graph to get a good upper bound for $ \mid f^{(4)}(x) \mid $.(g) Use $ S_{10} $ to approximate $ I $.(h) Use part (f) to estimate the error in part (g).(i) How does the actual error compare with the error estimate in part (h)?( j) How large should n be to guarantee that the size of the error in using $ S_n $ is less than 0.0001?

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a. $K=e$ or $K=2.8$b. $M_{10} \approx 7.954926518$c. $\left|E_{M}\right| \leq \frac{2.8(2 \pi-0)^{3}}{24 \cdot 10^{2}}=0.289391916$d. $I \approx 7.954926521$e. $3 \times 10^{-9}$f. $K=4 e$ or $K=10.9$g. $S_{10} \approx 7.953789422$h. $\left|E_{S}\right| \leq \frac{10.9(2 \pi-0)^{5}}{180 \cdot 10^{4}} \approx 0.059299814$i. $7.954926521-7.953789422 \approx 0.00114$j. $n \geq 50$ to ensure that $\left|I-S_{n}\right| \leq 0.0001$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

Oregon State University

Baylor University

University of Nottingham

Boston College

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

10:44

The trouble with the error…

27:20

30:53

17:24

(Continuation of Example $…

03:11

Use a CAS to estimate the …

04:27

a. Find a bound on the err…

Long problem, first, we should use the graph here's a graph. I take the second act of this function. It'S just a function, you have a problem and here's a plot of the absolute value, so you can see when x is equal to 0. The plot has its maximum. Actually is e is 2.71828 you that so here the upper bond is just 2.8 that use 1010. You should know how to compute 10 here's a formula you can find that 10 is just compute the formula and everything you know the you know the f x bar. You know, so you can compute. Men is just after a calculation. You can find a huntin, 7.945427 use part a to estimate. The error here is a function. Here is a formula here. The rebalance of m is k times b minus a cube divided by 24 times square. You know, you know, you know, and you know is just so. You can find a estimate. Error is 0.2894 here and is a butane integral capuality to approximate. You need a calculator now, the calculator that the answer is 7.9549 just calculate her is how does the actual error? What is the actual error? You need to know what errero is just i minus m right, that's actual error and compare with estimate. Here'S here is the estimated error just compare you can find that i i minus e, i minus m, is just quite smaller than is very small compared with you the graph to get a good upper band or a force. You need to take the directive force here. Let me show you how to use it. I used a soft, fair name mathematical. First, you first, you need to take the first active and just 2 into 4 point: here's a here's, a function and 1 to plot. The plot is copy that you can find that here is here is also the maximum of the function after calculation. K is equal to 4 is 10.9 acides 10.9, here 10.9, for how does actaeor compare with error to make? You need to confute. The error. To make right here is a formula. Is this formula because this use the simpsons rule right just compare the poland v to guarantee that the size of the error? As you can know, the k case for ye- and you know 4 times 2 pi it divided by 180 times square. This is smaller equal than 40111. You just use this to solve this equation. You can find that is greater than oh. There'S no answer. Let me check. Oh here's. 4. Actually, so you need to know what is astriction is greater than 4 times it is. Sequati. Is equal to 4 times e times, hi 2, the first iast so n is greater than 50 point. Here'S. The answer is quarter than 50 point. That'S all!

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