
Concept explainers
Evaluate the following
(a) Analytically;
(b) Single application of the trapezoidal rule;
(c) Multiple-application trapezoidal rule, with
(d) Single application of Simpson's 1/3 rule;
(e) Multiple-applicationSimpson's 1/3 rule, with
(f) Single application of Simpson's3/8 rule; and
(g) Multiple-application Simpson's rule, with
For each of the numerical estimates (b) through (g), determine the percent relative error based on (a).
(a)

To calculate: The value of the integral
Answer to Problem 2P
Solution:
Explanation of Solution
Given Information:
The integral,
Formula used:
Here,
Calculation:
Consider the integral,
The value of the integral is,
Therefore, the value of the integral is
(b)

To calculate: The value of the integral
Answer to Problem 2P
Solution:
Explanation of Solution
Given Information:
The integral,
The exact value of the integral
Formula used:
Single application version of Trapezoidal rule: If
Percentage error is,
Calculation:
Consider the integral,
Single application version of Trapezoidal rule is,
And
The value of the integral is,
And Percentage error is,
Therefore,
(c)

To calculate: The value of the integral
Answer to Problem 2P
Solution:
Explanation of Solution
Given Information:
The integral,
The exact value of the integral
Formula used:
Multiple application version of Trapezoidal rule: If
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Multiple application version of Trapezoidal rule is,
When
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Multiple application version of Trapezoidal rule is,
When
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
(d)

To calculate: The value of the integral
Answer to Problem 2P
Solution:
Explanation of Solution
Given Information:
The integral,
The exact value of the integral
Formula used:
Single application version of Simpson’s
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Single application version of Simpson’s
Here,
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
(e)

To calculate: The value of the integral
Answer to Problem 2P
Solution:
Explanation of Solution
Given Information:
The integral,
The exact value of the integral
Formula used:
Multiple application version of Simpson’s
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Multiple application version of Simpson’s
When
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
(f)

To calculate: The value of the integral
Answer to Problem 2P
Solution:
The value of the integral
Explanation of Solution
Given Information:
The integral,
The exact value of the integral
Formula used:
Single application version of Simpson’s
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Single application version of Simpson’s
Here,
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
(g)

To calculate: The value of the integral
Answer to Problem 2P
Solution:
The value of the integral
Explanation of Solution
Given Information:
The integral,
The exact value of the integral
Formula used:
Multiple application version of Simpson’s
Single application version of Simpson’s
Single application of Simpson’s
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Multiple application version of Simpson’s
Here,
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
Want to see more full solutions like this?
Chapter 21 Solutions
Numerical Methods for Engineers
Additional Math Textbook Solutions
College Algebra (Collegiate Math)
Elementary Statistics ( 3rd International Edition ) Isbn:9781260092561
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
Precalculus
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
Elementary Statistics: Picturing the World (7th Edition)
- Q/ Find the Laurent series of (2-3) cos around z = 1 2-1arrow_forward31.5. Let be the circle |+1| = 2 traversed twice in the clockwise direction. Evaluate dz (22 + 2)²arrow_forwardUsing FDF, BDF, and CDF, find the first derivative; 1. The distance x of a runner from a fixed point is measured (in meters) at an interval of half a second. The data obtained is: t 0 x 0 0.5 3.65 1.0 1.5 2.0 6.80 9.90 12.15 Use CDF to approximate the runner's velocity at times t = 0.5s and t = 1.5s 2. Using FDF, BDF, and CDF, find the first derivative of f(x)=x Inx for an input of 2 assuming a step size of 1. Calculate using Analytical Solution and Absolute Relative Error: = True Value - Approximate Value| x100 True Value 3. Given the data below where f(x) sin (3x), estimate f(1.5) using Langrage Interpolation. x 1 1.3 1.6 1.9 2.2 f(x) 0.14 -0.69 -0.99 -0.55 0.31 4. The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: 30 x = Loo (2000ln 140000 140000 - 2100 9.8t) dt Using the Trapezoidal Rule, n=2, find the distance covered. 5. Use Simpson's 1/3 and 3/8 Rule to approximate for sin x dx. Compare the results for n=4 and n=8arrow_forward
- 1. A Blue Whale's resting heart rate has period that happens to be approximately equal to 2π. A typical ECG of a whale's heartbeat over one period may be approximated by the function, f(x) = 0.005x4 2 0.005x³-0.364x² + 1.27x on the interval [0, 27]. Find an nth-order Fourier approximation to the Blue Whale's heartbeat, where n ≥ 3 is different from that used in any other posts on this topic, to generate a periodic function that can be used to model its heartbeat, and graph your result. Be sure to include your chosen value of n in your Subject Heading.arrow_forward7. The demand for a product, in dollars, is p = D(x) = 1000 -0.5 -0.0002x² 1 Find the consumer surplus when the sales level is 200. [Hints: Let pm be the market price when xm units of product are sold. Then the consumer surplus can be calculated by foam (D(x) — pm) dx]arrow_forward4. Find the general solution and the definite solution for the following differential equations: (a) +10y=15, y(0) = 0; (b) 2 + 4y = 6, y(0) =arrow_forward
- 5. Find the solution to each of the following by using an appropriate formula developed in the lecture slides: (a) + 3y = 2, y(0) = 4; (b) dy - 7y = 7, y(0) = 7; (c) 3d+6y= 5, y(0) = 0arrow_forward1. Evaluate the following improper integrals: (a) fe-rt dt; (b) fert dt; (c) fi da dxarrow_forward8. Given the rate of net investment I(t) = 9t¹/2, find the level of capital formation in (i) 16 years and (ii) between the 4th and the 8th years.arrow_forward
- 9. If the marginal revenue function of a firm in the production of output is MR = 40 - 10q² where q is the level of output, and total revenue is 120 at 3 units of output, find the total revenue function. [Hints: TR = √ MRdq]arrow_forward6. Solve the following first-order linear differential equations; if an initial condition is given, definitize the arbitrary constant: (a) 2 + 12y + 2et = 0, y(0) = /; (b) dy+y=tarrow_forward4. Let A = {a, b, c, d, e, f}, B = {e, f, g, h} and C = {a, e, h,i}. Let U = {a, b, c, d, e, f, g, h, i, j, k}. • Draw a Venn Diagram to describe the relationships between these sets Find (AB) NC • Find (AC) UB Find AUBUC • Find (BC) N (A - C)arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningIntermediate AlgebraAlgebraISBN:9781285195728Author:Jerome E. Kaufmann, Karen L. SchwittersPublisher:Cengage Learning




