Concept explainers
Compute the first-order central difference approximations of
(a)
(b)
(c)
(d)
(e)
Compare your results with the analytical solutions.
(a)
To calculate: The first order central difference approximations of
Answer to Problem 8P
Solution:
Analytic value of the first derivative is
Explanation of Solution
Given information:
The function,
The value of x,
Step size
Formula used:
Central difference approximation of
Here, h is step size and
Calculation:
Consider the function,
Differentiate the function with respect to x,
Now substitute
Thus, the analytic value of the first derivative of the function is
Again, consider the function,
Here,
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
Now, central difference approximation of
Substitute the values of
Therefore, the analytic value of the first derivative at
(b)
To calculate: The first order central difference approximations of
Answer to Problem 8P
Solution:
Analytic value of the first derivative is
Explanation of Solution
Given information:
The function,
The value of x,
Step size
Formula used:
Central difference approximation of
Here, h is step size and
Calculation:
Consider the function,
Differentiate the function with respect to x,
Now substitute
Thus, the analytic value of the first derivative of the function is
Again, consider the function,
Here,
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
Now, central difference approximation of
Substitute the values of
Therefore, the analytic value of the first derivative at
(c)
To calculate: The first order central difference approximations of
Answer to Problem 8P
Solution:
Analytic value of the first derivative is
Explanation of Solution
Given information:
The function,
The value of x,
Step size
Formula used:
Central difference approximation of
Here, h is step size and
Calculation:
Consider the function,
Differentiate the function with respect to x,
Now substitute
Thus, the analytic value of the first derivative of the function is
Again, consider the function,
Here,
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
Now, central difference approximation of
Substitute the values of
Therefore, the analytic value of the first derivative at
(d)
To calculate: The first order central difference approximations of
Answer to Problem 8P
Solution:
Analytic value of the first derivative is
Explanation of Solution
Given information:
The function,
The value of x,
Step size
Formula used:
Central difference approximation of
Here, h is step size and
Calculation:
Consider the function,
Differentiate the function with respect to x,
Now substitute
Thus, the analytic value of the first derivative of the function is
Again, consider the function,
Here,
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
Now, central difference approximation of
Substitute the values of
Therefore, the analytic value of the first derivative at
(e)
To calculate: The first order central difference approximations of
Answer to Problem 8P
Solution:
Analytic value of the first derivative is
Explanation of Solution
Given information:
The function,
The value of x,
Step size
Formula used:
Central difference approximation of
Here, h is step size and
Calculation:
Consider the function,
Differentiate the function with respect to x,
Now substitute
Thus, the analytic value of the first derivative of the function is
Again, consider the function,
Here,
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
Now, central difference approximation of
Substitute the values of
Therefore, the analytic value of the first derivative at
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