a.
To state: The approximate value of the integral
a.
Answer to Problem 19E
The resultant answer is 35.
Explanation of Solution
Given information:
The given integral is
Formula used: To approximate
Consider the given integral:
Now apply the Simpson’s Rule with
Now find
Now form a table as shown below:
x | |||
-1 | 1 | ||
0 | 0 | ||
1 | -1 | ||
2 | 4 | ||
3 | 21 |
Now substitute the value from the above table in the equation:
Therefore, the interval of convergence is
b.
To state: The exact value of the integral
b.
Answer to Problem 19E
The resultant exact value is 12 and the error
Explanation of Solution
Given information:
The given integral is
Consider the given integral:
If
This part of the fundamental theorem is called the integral evaluation theorem.
Now use the above formula to find the value of integral.
c.
To state: How a person predicted the value found in part (b) from knowing the error-bound formula.
c.
Answer to Problem 19E
The resultant answer is
Explanation of Solution
Given information:
The given integral is
The error
Here
Hence
d.
To state: The general statement about using Simpson’s Rule to approximate integrals of cubic polynomials.
d.
Answer to Problem 19E
The fourth derivate of the cubic polynomial always vanishes so error is zero. Hence, approximation given by Simpson’s rule is always equal to exact value of the integral.
Explanation of Solution
Given information:
The given integral is
The error
Here
A general cubic polynomial is:
Take fourth derivate both sides:
Since fourth derivate of the cubic polynomial always vanishes so error is zero. Hence, approximation given by Simpson’s rule is always equal to exact value of the integral.
Chapter 5 Solutions
Advanced Placement Calculus Graphical Numerical Algebraic Sixth Edition High School Binding Copyright 2020
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