1.
Assign the function \(f(x) = \dfrac{x}{x^2+4}\) using the assignment operator,
:=. Plot \(f(x)\) on the interval \([-5,10]\text{.}\)
Section 3.2 The Net Change Theorem
Subsection 3.2.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
Subsection 3.2.2 Introduction
The Net Change Theorem states that if a quantity \(Q = F(t)\) is a differentiable function on the interval \([a,b]\text{,}\) then
\begin{align*}
\int_{a}^b F'(t) \; dt \amp = F(b) - F(a)\\
\amp = \text{ net change in } Q \text{ over } [a,b]\text{.}
\end{align*}
In other words, the Net Change Theorem states that the definite integral of the rate of change of \(Q\) from \(a\) to \(b\) is given by the difference in the initial quantity and the final quantity.
We may also be interested in finding the total change of the quantity \(Q\text{,}\) given by the integral
\begin{equation*}
\int_{a}^{b} |F'(t)| dt\text{.}
\end{equation*}
In this case, all area is positive.
We will use Maple’s
ApproximateInt() command to help visualize the net change and total change of a function on an interval. In addition to the method=left and method=right parameters, we can also use method=upper and method=lower to ensure that our approximation is an overestimate or an underestimate, respectively.
Don’t forget that you will need to load the
Student[Calculus1] package to use the ApproximateInt() command. You can do this by typing with(Student[Calculus1]): at the start of your worksheet.
Exercises 3.2.3 Exercises
2.
Use the
ApproximateInt() command to estimate the net change of \(f(x)\) on the interval \([-5,10]\text{.}\) Use both method=upper and method=lower, with \(15\) partitions.
3.
In this exercise, you will determine the exact value of the net change of \(f(x)\) on the interval \([-5,10]\text{,}\) using a limit of Riemann sums as well as the convenient
Int() (or int()) command.
(a)
Use the
ApproximateInt() command with method=right and \(n\) partitions to give the Riemann sum for \(f(x)\) on the interval \([-5,10]\text{.}\) Then, use the limit() command to find the limit of this value as \(n\) goes to infinity.
Hint.
The
ApproximateInt() command will likely only output the net change in summation notation. You may need to force Maple to convert the summation to a closed form before evaluating the limit. To do this, follow up your ApproximateInt() output with a value(%) command before using the limit() command. The ditto operator % is a shortcut that reuses the most recent output.
(b)
Compute \(\displaystyle\int_{-5}^{10} f(x) \; dx\) by using the
Int() command. Verify that this value matches the limit of the Riemann sum in the previous part.
4.
Use the
ApproximateInt() command to estimate the total change of \(f(x)\) on the interval \([-5,10]\text{.}\) Use both method=upper and method=lower, with \(15\) partitions.
5.
In this exercise, you will determine the exact value of the total change of \(f(x)\) on the interval \([-5,10]\text{,}\) using a limit of Riemann sums as well as the convenient
Int() (or int()) command.
(a)
Use the
ApproximateInt() command with method=right and \(n\) partitions to give the Riemann sum for \(|f(x)|\) on the interval \([-5,10]\text{.}\) Then, use the limit() command to find the limit of this value as \(n\) goes to infinity.
Hint.
Once again, the
ApproximateInt() command will likely only output the total change in summation notation, using an absolute value. In this case, using the value(%) command might not give you a closed form of the summation!
As it turns out, evaluating total change as the limit of Riemann sums over the whole interval might not be the best way to go here. Don’t worry if you can’t get Maple to do what you want. You’ll use the
Int() command in a moment!
(b)
(c)
In a couple sentences, describe how you could go about evaluating the total change of \(f(x)\) on the interval \([-5,10]\) without needing to use the absolute value function.
