The Derivative of a Function

Section 1.9 The Derivative of a Function

Subsection 1.9.1 Recommended Tutorials

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

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Subsection 1.9.2 Introduction

In this activity, you will investigate the derivative of a function and use Maple’s powerful computational skills to simplify the process of finding a derivative.

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Exercises 1.9.3 Exercises

1.

Assign the function \(f(x) = \sqrt{9 - x}\) using the assignment operator, :=.

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(a)

You may evaluate the derivative of \(f(x)\) using the limit definition of the derivative. Use the limit() command to evaluate

\begin{equation*} f'(x) = \dlim{h}{0}\dfrac{f(x+h)-f(x)}{h}\text{.} \end{equation*}

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(b)

A convenient way to evaluate the derivative of an expression is by using the diff() command. Evaluate the derivative of \(f(x)\) using this method.

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(c)

When you have assigned something using proper function notation in Maple, you may quickly use ' notation to evaluate the derivative. Evaluate \(f'(x)\) using this method.

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2.

Molten lava can fill a chamber in the Earth’s crust before it builds up enough pressure to erupt. Suppose that the pressure of lava (in MPa) in a chamber is given by the function

\begin{equation*} P(t) = 0.47 t^2 { e}^{0.0035 t}\text{,} \end{equation*}

where \(t\) is the time in months. Start by assigning this function in Maple.

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(a)

Determine the rate of change of pressure as a function of time.

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Hint 1.

You may either use exp() for the exponential function or use the palettes toolbar. Don’t forget to use \(t\) as your variable instead of \(x\text{!}\)

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Hint 2.

If you have assigned the pressure of lava as a function, you can use ' notation. You can always use diff() instead, so long as you indicate the correct variable (\(t\) in this case).

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(b)

Determine the rate of change of pressure at \(t=30\) months. If an eruption is likely to occur when the rate of pressure is above 20 MPa/month, is an eruption likely at this time? Use a new paragraph to state your answer.

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Hint.

If you are already using ' notation, then you can simply evaluate \(P'(30)\text{.}\) If you are using diff() instead, then you will need to make use of the subs() command to substitute \(t=30\) into your derivative expression.

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3.

A toy rocket is fired straight upward, and its height (in metres) is given by

\begin{equation*} h(t) = t + 10 - \sqrt{2t^2 + 100} \qquad 0 \leq t \leq 20\text{,} \end{equation*}

where \(t\) is the time in seconds. Begin by assigning this expression in Maple.

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(a)

Plot a graph of the height of the rocket over the specified interval of time.

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Hint.

When specifying an interval for your horizontal axis, make sure that you use \(t\) as your variable instead of \(x\text{.}\)

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(b)

Determine the vertical velocity of the rocket as a function of time.

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Hint.

Recall that velocity is the rate of change of position of an object.

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(c)

Use the velocity function to determine when the rocket reaches its maximum height.

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Hint.

At the maximum height, the rocket’s vertical velocity should equal zero. Can you solve for the time at which this occurs?

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(d)

What is the maximum height that the rocket attains?

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Hint.

An example of finding minimum and maximum values on a closed interval can be found in Example 12.2.

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4.

The higher derivatives of \(\sin(x)\) and \(\cos(x)\) follow a predictable pattern. In this exercise, you will look for a pattern in the derivatives of the function

\begin{equation*} g(x)=\sin(2\pi^2 x)\text{.} \end{equation*}

The derivatives of \(g(x)\) will require use of the chain rule, which makes calculations more complicated. However, there is still a predictable pattern of higher derivatives.

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Aside

(a)

Assign the function \(g(x)=\sin(2\pi^2 x)\text{.}\)

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Hint 1.

To type the mathematical constant \(\pi = 3.14\dots\text{,}\) be sure to use Pi.

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Hint 2.

Don’t forget to include multiplication between \(\pi^2\) and \(x\text{.}\)

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(b)

Use Maple to find the first, second, third, and fourth derivatives of \(g(x)\text{.}\) You should notice a pattern. Try to describe this pattern in a paragraph.

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(c)

Using this pattern, try to predict the 77th derivative of \(g(x)\text{.}\) Then, evaluate the 77th derivative directly using a single diff() command.

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(d)

(Optional) Use the examples in Chapter 18 to write a loop that will output the first \(100\) derivatives of \(g(x)\text{.}\)

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