(a)
Approximate the value of integral using Trapezoidal rule with
(a)
Answer to Problem 5E
With
Explanation of Solution
Given information:
Since we are required to use the Trapezoidal rule, first we need to find the height.
Function values:
Now,
Setup the equation:
Plugin the values into the equation:
(b)
Whether the approximation is an overestimate or an under − estimate.
(b)
Answer to Problem 5E
The approximation is an under − estimate.
Explanation of Solution
Given information:
First derivative of the function:
Second derivative of the function:
We know that
Second derivative gives the concavity.
If the second derivative is greater than 0,
It’s an overestimate.
If the second derivative is less than 0,
It’s an underestimate.
Note that
The value of second derivative is
Therefore,
The approximation is an under − estimate.
(c)
Integral’s exact value to verify the answer.
(c)
Answer to Problem 5E
The exact value of integral is approx. 5.33 which is not same as using the Trapezoidal rule with
Explanation of Solution
Given information:
Using FTC (Fundamental Theorem of Calculus):
Therefore,
Integral’s exact value is approx. 5.33 which is not same as the value of integral obtained in Part (a).
Chapter 6 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
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