Applications of the Fundamental Theorem of Calculus

Section 3.5 Applications of the Fundamental Theorem of Calculus

Subsection 3.5.1 Recommended Tutorials

Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:

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Subsection 3.5.2 Introduction

In this activity, you will need to make use of the Fundamental Theorem of Calculus in order to solve two applied problems. In particular, you will need to use the result

\begin{equation*} \frac{d}{dx} \int_a^x f(t) \, dt = f(x)\text{.} \end{equation*}

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Exercises 3.5.3 Exercises

1.

In this exercise, your goal is to evaluate a challenging limit problem:

\begin{equation*} \displaystyle\lim_{x \rightarrow 0} \dfrac{\int_{0}^x \sin(t^2) \; dt}{x^3}\text{.} \end{equation*}

Start by assigning the function \(s(x) = \dfrac{\int_{0}^x \sin(t^2) \; dt}{x^3}\) using the assignment operator, :=.

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Aside

(a)

Try visualizing the problem by plotting the graph of \(s(x)\) over the interval \([-1,1]\text{.}\) See if you can estimate \(\displaystyle\lim_{x\rightarrow0}s(x)\) from the plot.

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(b)

The integral function in the numerator is what makes this limit problem so challenging. Try evaluating \(\int_{0}^x \sin(t^2) \; dt\) using the int() command or Int() and value(%) commands. Notice how Maple evaluates the integral, but not in terms of elementary functions.

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Hint.

There is no easy integration technique for this integral. You’ll notice that Maple evaluates the integral in terms of the Fresnel S function, a transcendental function named after Augustin-Jean Fresnel.

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(c)

Luckily, you should not need to evaluate this integral in order to evaluate the limit. Instead, you may apply l’Hôpital’s rule, so long as the limit is indeterminate of the form \(0/0\) or \(\infty/\infty\text{.}\) Which of these two indeterminate forms is applicable?

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(d)

Applying l’Hôpital’s rule to the limit allows you to differentiate the numerator and denominator and obtain a new expression for the limit as \(x \to 0\text{.}\) Use the diff() command to differentiate the numerator and denominator of \(s(x)\) and see how the Fundamental Theorem of Calculus is applied.

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Hint.

You can differentiate the numerator of \(s(x)\) by using the numer() function:

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> diff(numer(s(x)), x);

Similarly, you can differentiate the denominator of \(s(x)\) using the denom() function.

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Try making a new fraction using both of these functions together.

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(e)

You should have noticed that the new limit after applying l’Hôpital’s rule is indeterminate of the form \(0/0\text{.}\) Try applying l’Hôpital’s rule again, this time finding the second derivatives of the numerator and denominator.

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Hint.

You can find the second derivative of the numerator of \(s(x)\) using

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> diff(numer(s(x)), x, x);

and switching to denom() for the denominator.

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(f)

After applying l’Hôpital’s rule twice, you should be able to evaluate the limit. Confirm your answer by using the limit() command to evaluate \(\displaystyle\lim_{x\rightarrow0}s(x)\text{.}\)

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2.

In this exercise, your goal is to solve for a function \(f\) and for a number \(a\) that satisfy the equation

\begin{equation} 6 + \displaystyle\int_{a}^x \dfrac{f(t)}{t^2} dt = 2\sqrt{x}\text{.}\tag{3.4} \end{equation}

Start by assigning equation (3.4) to a name of your choice, such as eqn.

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Aside

(a)

To solve for the function \(f\text{,}\) you can differentiate both sides of the equation and apply the FTC. Use the diff() command on your equation to differentiate both sides. Then, solve for \(f(x)\) using the solve() command.

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(b)

Substitute the solution for \(f\) into equation (3.4) and assign it to a new name, such as eqn2.

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Hint.

Make sure that you are using \(t\) as the variable in the integrand when substituting your solution for \(f\text{.}\)

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(c)

Evaluate the integral in the equation and solve for \(a\) using the solve() command.

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