3.5: Derivatives of Polynomials, Sum and Constant Factor Rules

Note

We begin a common strategy to find derivatives of functions:

• Use Definition 3.2.2 to find the derivative of a few elementary functions (such as \(F(t) = C, F(t) = t\), and \(F(t) = t^n\) ). We call these formulas primary formulas.
• Use Definition 3.2.2 to prove some rules about derivatives of combinations of functions, called combination rules.
• Use the combination rules and primary formulas to compute derivatives of more complex functions.

We are using the symbol t for the independent variable. We could as well use x as in previous sections, but time is a common independent variable in biological models and we wish to use both symbols.

The following are derivative formulas for this section.

Primary formulas: First Set

If \(C\) is a number and \(n\) is a positive integer and \(u, v\), and \(F\) are functions with a common domain, \(D\), then

\[\begin{align}
\text{If for all } t, \quad &F(t)=C \quad &\text{then} \quad &F^{\prime}(t)=0 \quad &\textbf{Constant Rule} \\
\text{If for all } t, \quad &F(t)=t \quad &\text{then} \quad &F^{\prime}(t)=1 \quad &\textbf{t Rule} \\
\text{If for all } t, \quad &F(t)=t^{n} \quad &\text{then} \quad &F^{\prime}(t)=n t^{n-1} \quad &\mathbf{{t}^{n} Rule}
\end{align}\]

Combination Rules: First Set

\[\begin{align}
\text{If for all t in D}, \quad & F(t) = u(t) + v(t) \quad &\text{then} \quad &F^{\prime}(t)=u^{\prime}(t)+v^{\prime}(t) \quad &\textbf{Sum Rule} \\
\text{If for all t in D} , \quad &F(t) = C u(t) \quad &\text{then} \quad &F^{\prime}(t)=C u^{\prime}(t) \quad &\textbf{Constant}\\
&&&&\textbf{Factor Rule} \\
\end{align}\]

We use a notation that simplifies the statements of the rules. An example is

\[\text { If } F(t)=t^{4} \text { then } F^{\prime}(t)=4 t^{3} \quad \text { is shortened to } \quad\left[t^{4}\right]^{\prime}=4 t^{3} \text { . }\]

More generally

\[\text { If } \quad F(t)=E(t) \text { then } \quad F^{\prime}(t)=H(t) \quad \text { is shortened to } \quad[E(t)]^{\prime}=H(t) \text { . }\]

Alternatively, we may use Leibnitz’ notation and write

\[\frac{d}{d t}[E(t)]=H(t)\]

The first set of rules may be written,

Derivative Rules: First Set

\[\begin{align}\\
{[C]^{\prime}} & =0 & & \textbf {Constant Rule } \label{3.27}\\
{[t]^{\prime}} & =1 & & \textbf {t Rule } \label{3.28}\\
{\left[t^{n}\right]^{\prime}} & =n t^{n-1} & & \mathbf{t^n} \textbf { Rule } \label{3.29}\\
{[u(t)+v(t)]^{\prime}} & =u^{\prime}(t)+v^{\prime}(t) & & \textbf {Sum Rule } \label{3.30}\\
{[C F(t)]^{\prime}} & =C F^{\prime}(t) & & \textbf {Constant Factor Rule } \label{3.31}\\
\end{align}\]

The short forms depend on your recognizing that \(C\) is a constant, \(t\) is an independent variable, \(n\) is a positive integer, and that \(u\) and \(v\) are functions with a common domain, and that \(F\) is a function.

The distinction between \([C]^{\prime}=0\) and \([t]^{\prime} = 1\) is that the symbol \([\text{ }] ^{\prime}\) means rate of change as \(t\) changes, sometimes said to be derivative with respect to \(t\). \(C\) denotes a number that does not change with \(t\) and therefore its rate of change as \(t\) changes is zero. The Leibnitz notation, \(\frac{d}{dt}\), is read, 'derivative with respect to t', and gives a better distinction between these two formulas.

Explore 3.5.1 Write short forms of the first set of derivative rules using the Leibnitz notation, \(\frac{d}{dt}\).

Proofs of the Derivative Rules: First Set. You have sufficient experience that you can prove each of these rules using the Definition of Derivative 3.3.3. We prove the \(t^n\) Rule and the Sum Rule and leave the others for you in Exercise 3.5.4.

Explore 3.5.2 The steps of the arguments for the Sum Rule and the Constant Factor Rule will be made apparent if you use Definition of Derivative 3.3.3 to compute the derivatives of

\[P(t)=t^{2}+t^{3} \quad \text { and } \quad P(t)=4 t^{2}\]

 

\(t^n\) Rule. Suppose \(n\) is a positive integer and a function \(F\) is defined by

\[F(t)=t^{n} \quad \text { for all numbers } t . \quad \text { Then } \quad F^{\prime}(t)=n t^{n-1}\]

Proof.

\[\begin{aligned}
F^{\prime}(t)&=\lim _{b \rightarrow t} \frac{F(b)-F(t)}{b-t}\\
&=\lim _{b \rightarrow t} \frac{b^{n}-t^{n}}{b-t} \\
&=\lim _{b \rightarrow t} \frac{(b-t)\left(b^{n-1}+b^{n-2} t+\cdots+b t^{n-2}+t^{n-1}\right)}{b-t} \\
&=\lim _{b \rightarrow t}\left(b^{n-1}+b^{n-2} t+\cdots+b t^{n-2}+t^{n-1}\right) \\
&=\underbrace{t^{n-1}+t^{n-1}+\cdots+t^{n-1}+t^{n-1}}_{n \text { terms }} \\
&=n t^{n-1}
\end{aligned}\]

We have stated and proved the \(t^n\) Rule assuming that \(n\) is a positive integer. It is also true for \(n\) a negative integer when \(t \neq 0\) (Exercise 3.5.5). When \(t\) is restricted to being greater than zero, the \(t^n\) rule is valid for every number \(n\) (integer, rational, and irrational). You will prove it for rational numbers \(\frac{p}{q}\), \(p\) and \(q\) integers in the Chapter 4, Exercise 4.3.4. Meanwhile, we encourage you to use formulas such as

\[\left[t^{-5}\right]^{\prime}=-5 t^{-5-1}=-5 t^{-6}, \quad\left[t^{5 / 3}\right]^{\prime}=\frac{5}{3} t^{(5 / 3)-1}=\frac{5}{3} t^{2 / 3}, \quad \text { and even } \quad\left[t^{\pi}\right]^{\prime}=\pi t^{\pi-1} .\]

 

Sum Rule. Suppose you and your lab partner are growing two populations of Vibrio natriegens, one growing at the rate of \(4.32 \times 10^5\) cells per minute and the other growing at the rate of \(2.19 \times 10^5\) cells per minute. It is a fairly easy conclusion that the growth rate of all V. natriegens in the two populations is \(4.32 \times 10^5\) plus \(2.19 \times 10^5\) = \(6.51 \times 10^5\) cells per minute.

It is your experience that rates are additive, and we will include a proof. The proof is needed to distinguish the intuitive for addition from the non-intuitive for multiplication. Many students ‘intuitively think’ that rates are multiplicative, and they are not!

Proof. Suppose \(u\) and \(v\) are functions with a common domain, \(D\), and for all \(t\) in \(D\), \(F(t) = u(t) + v(t)\).

\[\begin{aligned}
F^{\prime}(t) &=\lim _{b \rightarrow t} \frac{F(b)-F(t)}{b-t} \\
&=\lim _{b \rightarrow t} \frac{u(b)+v(b)-(u(t)+v(t))}{b-t} \\
&=\lim _{b \rightarrow t} \frac{u(b)-u(t)+v(b)-v(t)}{b-t} \\
&=\lim _{b \rightarrow t} \frac{u(b)-u(t)}{b-t}+\frac{v(b)-v(t)}{b-t} \\
&=u^{\prime}(t)+v^{\prime}(t)
\end{aligned}\]

The sum rule has a companion, the difference rule,

\[[u(t)-v(t)]^{\prime}=u^{\prime}(t)-v^{\prime}(t)\]

that we also call the sum rule. Furthermore,

\[[u(t)+v(t)+w(t)]^{\prime}=[u(t)+v(t)]^{\prime}+w^{\prime}(t)=u^{\prime}(t)+v^{\prime}(t)+w^{\prime}(t)\]

It can be shown by induction that

\[\left[u_{1}(t)+u_{2}(t)+\cdots+u_{n}(t)\right]^{\prime}=u_{1}^{\prime}(t)+u_{2}^{\prime}(t)+\cdots+u_{n}^{\prime}(t)\]

'Sum Rule' encompasses all of these possibilities.

Use of the derivative rules. Using the first set of derivative rules, we can compute the derivative of any polynomial without explicit reference to the Definition of Derivative 3.3.3, as the following example illustrates.

Example 3.5.1 Let \(P(t) = 3t^{4} − 5t^{2} + +2t + 7\). Then

\[\begin{aligned}
P^{\prime}(t) &=[P(t)]^{\prime} & \text {Notation shift} \\
&=\left[3 t^{4}-5 t^{2}+2 t+7\right]^{\prime} & \text {Definition of } P \\
&=\left[3 t^{4}\right]^{\prime}-\left[5 t^{2}\right]^{\prime}+[2 t]^{\prime}[7]^{\prime} & \text{Sum Rule}\\
&=3\left[t^{4}\right]^{\prime}-5\left[t^{2}\right]^{\prime}+2[t]^{\prime}[7]^{\prime} & \text {Constant Factor Rule} \\
&=3 \times 4 t^{3}-5 \times 2 t^{1}+2[t]^{\prime}[7]^{\prime} & t^{n} \text { Rule} \\
&=12 t^{3}-10 t+2 \times 1+[7]^{\prime} & \text {t Rule}\\
&=12 t^{3}-10 t+2+0 & \text{Constant Rule}\\
\end{aligned}\]

Example 3.5.2 It is useful to write some fractions in forms so that the Constant Factor Rule obviously applies. For example

\[P(t)=\frac{t^{3}}{7}=\frac{1}{7} t^{3}\]

and to compute \(P ^{\prime}\) you may write

\[P^{\prime}(t)=\left[\frac{t^{3}}{7}\right]^{\prime}=\left[\frac{1}{7} t^{3}\right]^{\prime}=\frac{1}{7}\left[t^{3}\right]^{\prime}=\frac{1}{7} 3 t^{2}=\frac{3}{7} t^{2}\]

Similarly

\[P(t)=\frac{8}{3 t}=\frac{8}{3} \frac{1}{t}\]

and to compute \(P ^{\prime}\) you may write

\[P^{\prime}(t)=\left[\frac{8}{3 t}\right]^{\prime}=\left[\frac{8}{3} \frac{1}{t}\right]^{\prime}=\frac{8}{3}\left[\frac{1}{t}\right]^{\prime}=\frac{8}{3}\left[t^{-1}\right]^{\prime}=\frac{8}{3}\left((-1) t^{-1-1}\right)=-\frac{8}{3}\left(\frac{1}{t^{2}}\right)\]

Explore 3.5.3 Let \(P(t)=1+2 t^{3} / 5\). Cite the formulas that justify the steps (i) − (iv).

\[\begin{aligned}
P^{\prime}(t) &=\left[1+2 \frac{t^{3}}{5}\right]^{\prime} &\\
&=[1]^{\prime}+\left[2 \frac{t^{3}}{5}\right]^{\prime} & (i)\\
&=0+\left[2 \frac{t^{3}}{5}\right]^{\prime} & (ii)\\
&=0+\left[\frac{2}{5} t^{3}\right]^{\prime} &\\
&=0+\frac{2}{5}\left[t^{3}\right]^{\prime} & (iii)\\
&=0+\frac{2}{5} \times 3 t^{2} & (iv)\\
&=\frac{6}{5} t^{2} &\\
\end{aligned}\]

Example 3.5.3 The derivative of a quadratic function is a linear function.

Solution. Suppose \(P(t) = at^{2} + bt + c\) where \(a, b\), and \(c\) are constants. Then

\[\left.\begin{array}{rl}
P^{\prime}(t) & =\left[a t^{2}+b t+c\right]^{\prime} & (i)\\
& =\left[a t^{2}\right]^{\prime}+[b t]^{\prime}+[c]^{\prime} & (ii)\\
& =a\left[t^{2}\right]^{\prime}+b[t]^{\prime}+[c]^{\prime} & (iii)\\
& =a \text{ } 2 t+b \times 1+[c]^{\prime} & (iv)\\
& =2 a t+b+0 & (v)\\
\end{array}\right\} \label{3.32}\]

We see that \(P^{\prime}(t)=2 a t+b\) which is a linear function.

 

3.5.1 Velocity as a derivative.

If \(P(t)\) denotes the position of a particle along an axis at time \(t\), then for any time interval [a, b],

\[\frac{P(b)-P(a)}{b-a}\]

is the average velocity of the particle during the time interval [a, b]. The rate of change of \(P\) at \(t = a\),

\[\lim _{b \rightarrow a} \frac{P(b)-P(a)}{b-a}\]

is the velocity of the particle at time \(t = a\).

Example 3.5.4 In baseball, a ‘pop fly’ is hit and the ball leaves the bat traveling vertically at 30 meters per second. How high will the ball go and how much time does the catcher have to get in position to catch it?

Solution. Using a formula from Section 3.6.1, the ball will be at a height \(s(t) = −4.9t^{2} + 30t\) meters \(t\) seconds after it is released, where \(s(t)\) is the height above the point of impact with the bat. The velocity, \(v(t) = s ^{\prime} (t)\) is

\[[s(t)]^{\prime}=\left[-4.9 t^{2}+30 t\right]^{\prime}=-4.9 \cdot 2 t+30 \cdot 1=-9.8 t+30\]

The ball will be at its highest position when the velocity \(v(t) = s ^{\prime} (t) = 0\) (which implies that the ball is not moving and identifies the time at which the ball is at its highest point, is not going up and is not going down).

\[s^{\prime}(t)=0 \quad \text { implies } \quad-9.8 t+30=0, \quad \text { or } \quad t=\frac{30}{9.8} \approx 3.1 \text { seconds. }\]

The height of the ball at \(t = 3.1\) seconds is

\[s(3.1)=-4.9(3.1)^{2}+30 \times 3.1 \approx 45.9 \text { meters }\]

Thus in about 3.1 seconds the ball reaches a height of about 45.9 meters. The catcher will have about 6 seconds to position to catch the ball. Furthermore, at time \(t = 6.2\)

\[s^{\prime}(6.2)=-9.8 \times 6.2+30=-30.76 \quad \text { meters/second }\]

The velocity of the falling ball is \(\approx −30.76\) m/s when the catcher catches it. Its magnitude will be exactly 30 m/s, the speed at which it left the bat. Why not 30.76?

 

3.5.2 Local Maxima and Local Minima.

Example 3.5.4 illustrates a useful technique that will be expanded in the next chapter: if one is seeking the high point of a graph, it is useful to examine the points at which the derivative of the related function is zero and the tangent is horizontal. However, tangents at local minima are also horizontal, so that knowing the location of a horizontal tangent does not insure that the location is a local maximum – it might be a local minimum or it may be neither a local minimum nor a local maximum!

Shown in Figure 3.21A is the graph of a function and two horizontal tangents to the graph. The horizontal tangent at A marks a local maximum of the graph, the tangent at B marks a local minimum of the graph. A is a ‘local’ maximum because there is an open interval, \(\alpha\), of the domain surrounding the x-coordinate of A and if P is a point of the graph with x-coordinate in \(\alpha\), then P is not above A. But note that there are points of the graph above A. In Figure 3.5.1B is the graph of \(P(t) = t^3\) which has a horizontal tangent at (0, 0). The horizontal tangent at (0, 0), however, signals neither a local maximum nor a local minimum.

Figure \(\PageIndex{1}\): A. Graph with a local maximum at A and a local minimum at B. B. Graph of \(P(t) = t^3\) that has a horizontal tangent at (0, 0); (0, 0) is neither a local maximum nor a local minimum of P.

Example 3.5.5 A farmer’s barn is 60 feet long on one side. He has 100 feet of fence and wishes to build a rectangular pen along that side of his barn. What should be the dimensions of the pen to maximize the area?

A diagram of a barn and fence with some important labels, \(L\) and \(W\), is shown in Figure 3.5.2. Because there are 100 feet of fence,

\[2 * W+L=100\]

The area, A, is

\[A=L W\]

Figure \(\PageIndex{2}\): Diagram of a barn with adjacent pen bounded by a fence of length 100 feet.

Because \(2W + L = 100\),

\[L=100-2 W\]

and

\[A=L W=(100-2 W) W\]

or

\[A=100 W-2 W^{2}\]

The question becomes now, for what value of W will A be the largest. The graph of A vs W is a parabola with its highest point at the vertex. The tangent to the parabola at the vertex is horizontal, and we find a value of W for which \(A^{\prime} (W) = 0\).

\[\begin{array} \\
A^{\prime}(W) &=\left[100 W-2 W^{2}\right]^{\prime} & (i)\\
&=[100 W]^{\prime}-\left[2 W^{2}\right]^{\prime} & (ii)\\
&=100[W]^{\prime}-2\left[W^{2}\right]^{\prime} & (iii)\\
&=100 \times 1-2 \times 2 W & (iv) \\
\end{array} \label{3.33}\]

The optimum dimensions, \(W\) and \(L\), are found by setting \(A^{\prime} (W) = 0\), so that

\[\begin{aligned}
A^{\prime}(W) &=100-4 W=0 \\
W &=25 \\
L &=100-2 W=50
\end{aligned}\]

Thus the farmer should build a 25 by 50 foot pen.

Example 3.5.6 This problem is written on the assumption, to our knowledge untested, that spider webs have an optimum size. Seldom are they so small as 1 cm in diameter and seldom are they so large as 2 m in diameter. If they are one cm in diameter, there is a low probability of catching a flying insect; if they are 2 m in diameter they require extra strength to withstand wind and rain. We will examine circular webs, for convenience, and determine the optimum diameter for a web so that it will catch enough insects and not fall down.

Solution. Assume a circular spider web of diameter, \(d\). It is reasonable to assume that the amount of food gathered by the web is proportional to the area, \(A\), of the web. Because \(A = \pi d^{2}/4\), the amount of food gather is proportional to \(d^2\). We also assume that the energy required to build and maintain a web of area \(A\) is proportional to \(d^3\). (The basic assumption is that the work to build a square centimeter of web increases as the total web area increases because of the need to have stronger fibers. If, for example, the area of the fiber cross-section increases linearly with \(A\) and the mesh of the web is constant, the mass of the web increases as \(d^3\).)

With these assumptions, the net energy, \(E\), available to the spider is of the form

\[\begin{aligned}
E &=\text { Energy from insects caught }-\text { Energy expended building the web } \\
&=k_{1} d^{2}-k_{2} d^{3}
\end{aligned}\]

where \(d\) is measured in centimeters and \(k_1\) and \(k_2\) are proportionality constants.

For illustration we will assume that \(k_{1} = 0.01\) and \(k_2 = 0.0001\). A graph of \(E(d) = 0.01d^{2} − 0.0001d^{3}\) is shown in Figure 3.5.3 where it can be seen that there are two points, A and B, at which the graph has horizontal tangents. We find where the derivative is zero to locate A and B.

\[\begin{array} \\
E^{\prime}(d) &=\left[0.01 d^{2}-0.0001 d^{3}\right]^{\prime} & (i)\\
&=\left[0.01 d^{2}\right]^{\prime}-\left[0.0001 d^{3}\right]^{\prime} & (ii)\\
&=0.01\left[d^{2}\right]^{\prime}-0.0001\left[d^{3}\right]^{\prime} & (iii) \\
&=0.01 \times 2 d-0.0001 \times 3 d^{2} & (iv) \\
&=0.02 d-0.0003 d^{2} & \\
\end{array} \label{3.34}\]

Figure \(\PageIndex{3}\): Graph of \(E(d) = 0.01d^{2} − 0.0001d^{3}\) representative of the energy gain from a spider web of diameter \(d\) cm. There is a local maximum at A and a local minimum at B.

Then \(E ^{\prime} (d) = 0\) yields

\[\begin{aligned}
0.02 d-0.0003 d^{2} &=0 \\
d(0.02-0.0003 d) &=0 \\
d &=0 \quad \text { or } \quad d=0.02 / 0.0003 \doteq 66.7 \quad \mathrm{~cm}
\end{aligned}\]

The value \(d = 0\) locates the local minimum at B and has an obvious interpretation: if there is no web there is no energy gain. At \(d\) = 66.7 cm, \(E(d) = 14.8\) (units unspecified) suggests that a positive net energy will accrue with a web of diameter 66.7 cm and that weaving a web of 66.7 cm diameter is the optimum strategy for the spider. Note that if our model and its parameters are correct we have determined in a rather short time what it took spiders many generations to work out. At the very least we could have moved the spiders ahead several generations with our model.

 

Exercises for Section 3.5, Derivatives of Polynomials, Sum and Constant Factor Rules.

Exercise 3.5.1 Use Definition of the Derivative 3.22 to compute \(P ^{\prime} (t)\) for

1. \(P(t)=1+t^{2}\)
2. \(P(t)=t-t^{2}\)
3. \(P(t)=t^{2}-t\)
4. \(P(t)=5 t^{2}\)
5. \(P(t)=5 \times 3\)
6. \(P(t)=1+5 t^{2}\)
7. \(P(t)=1+t+t^{2}\)
8. \(P(t)=2 t+3 t^{2}\)
9. \(P(t)=5+3 t-2 t^{2}\)

Exercise 3.5.2 Suppose \(u\) and \(v\) are functions with a common domain and \(P = u − v\). Write \(P(t) = u(t) + (−1) v(t)\) and use the Sum and Constant Factor rules to show that \(P 0 (t) = u ^{\prime} (t) − v ^{\prime} (t)\).

Exercise 3.5.3 Provide reasons for the steps (i) − (v) in Equations \ref{3.32} use to show that the derivatives of quadratic functions are linear functions.

Exercise 3.5.4

1. Prove Equation \ref{3.27}, \([ C ] ^{\prime} = 0\).
2. Prove Equation \ref{3.28}, \([ t ] ^{\prime} = 1\).
3. Prove Equation \ref{3.31}, \([ C F(t) ]^{\prime} = C [ F(t) ]^{\prime}\).

Exercise 3.5.5 Suppose m is a positive integer and \(u(t) = t ^{−m} = 1/t^{m}\) for \(t \neq 0\). Show that \(u ^{\prime} (t) = −mt^{−m−1}\) , thus proving the \(t^n\) rule for negative integers. Begin your argument with

\[\begin{aligned}
u^{\prime}(t) &=\lim _{b \rightarrow t} \frac{\frac{1}{b^{m}}-\frac{1}{t^{m}}}{b-t} \\
&=\lim _{b \rightarrow t} \frac{t^{m}-b^{m}}{b^{m} t^{m}} \times \frac{1}{b-t}
\end{aligned}\]

Exercise 3.5.6 Provide reasons for the steps (i) − (iv) in Equations \ref{3.33} used to find the optimum dimensions of a lot adjacent to a barn.

Exercise 3.5.7 A farmer’s barn is 60 feet long on one side. He has 120 feet of fence and wishes to build a rectangular pen along that side of his barn. What should be the dimensions of the pen to maximize the the area of the pen?

Exercise 3.5.8 A farmer’s barn is 60 feet long on one side. He has 150 feet of fence and wishes to build two adjacent rectangular pens of equal area along that side of his barn. What should be the arrangement and dimensions of the pen to maximize the sum of the areas of the two pens?

Exercise 3.5.9 A farmer’s barn is 60 feet long on one side. He has 150 feet of fence and wishes to build two adjacent rectangular pens of equal area along that side of his barn. What should be the arrangement and dimensions of the pen to maximize the sum of the areas of the two pens?

Exercise 3.5.10 A farmer’s barn is 60 feet long on one side. He has 280 feet of fence and wishes to build two adjacent rectangular pens of equal area along that side of his barn. What should be the arrangement and dimensions of the pen to maximize the sum of the areas of the two pens?

Exercise 3.5.11 A farmer’s barn is 60 feet long on one side. He wishes to build a rectangular pen of area 800 square feet along that side of his barn. What should be the dimension of the pen to minimize the amount of fence used?

Exercise 3.5.12 Show that the derivatives of cubic functions are quadratic functions.

Exercise 3.5.13 Probably baseball statistics should be discussed in British units rather than metric units. Professional pitchers throw fast balls in the range of 90+ miles per hour. Suppose the pop fly ball leaves the bat traveling 60 miles per hour (88 feet/sec), in which case the height of the ball in feet will be \(s(t) = −16t^{2} + 88t\) feet above the bat, t seconds after the batter hits the ball. How high will the ball go, and how long will the catcher have to position to catch it? How fast is the ball falling when the catcher catches it?

Exercise 3.5.14 A squirrel falls from a tree from a height of 10 meters above the ground. At time t seconds after it slips from the tree, the squirrel is a distance \(s(t) = 10 − 4.9t^2\) meters above the ground. How fast is the squirrel falling when it hits the ground?

Exercise 3.5.15 What is the optimum radius of the trachea when coughing? The objective is for the flow of air to create a strong force outward in the throat to clear it.

For this problem you should perform the following experiment.

Hold your hand about 10 cm from your mouth and blow on it (a) with your lips compressed almost closed but with a small stream of air escaping, (b) with your mouth wide open, and (c) with your lips adjusted to create the largest force on your hand. With (a) your lips almost closed there is a high pressure causing rapid air flow but a small stream of air and little force. With (b) your mouth wide open there is a large stream of air but with little pressure so that air flow is slow. The largest force (c) is created with an intermediate opening of your lips where there is a notable pressure and rapid flow of substantial volume of air.

Let R be the normal radius of the trachea and \(r < R\) be the tracheal radius when coughing. Assume that the velocity of air flow through the trachea is proportional to pressure difference across the trachea and that the pressure difference is proportional to \(R - r\), the constriction of the trachea. Assume that the mass of air flow is proportional to the area of the trachea (\(\pi r^2\)). Finally, the momentum, \(M\), of the air flow is mass times velocity:

\[M=k r^{2}(R-r)\]

What value of \(r\), \(0 < r < R\), maximizes \(M\)?

Exercise 3.5.16 Cite formulas that justify the steps (i) − (iv) in Equations \ref{3.34} for the analysis of the spider web.

Exercise 3.5.17 Let \(P(t) = −3 + 5 t − 2 t^2\). Cite the formulas that justify steps (i) − (vi) below:

\[\begin{aligned}
P^{\prime}(t) &=\left[-3+5 t+(-2) t^{2}\right]^{\prime} & (i)\\
&=[-3]^{\prime}+[5 t]^{\prime}+\left[(-2) t^{2}\right]^{\prime} & (ii)\\
&=0+[5 \times t]^{\prime}+\left[(-2) t^{2}\right]^{\prime} & (iii)\\
&=0+5[t]^{\prime}+(-2)\left[t^{2}\right]^{\prime} & (iv)\\
&=0+5 \times 1+(-2)\left[t^{2}\right]^{\prime} & (v)\\
&=0+5 \times 1+(-2) \times 2 t & (vi)\\
&=5-4 t
\end{aligned}\]

Exercise 3.5.18 Compute the derivatives of the following polynomials as in the previous exercises. Use only one rule for each step written, and write the name of the rule used for each step.

1. \(P(t)=15 t^{2}-32 t^{6}\)
2. \(P(t)=1+t+t^{2}+t^{3}\)
3. \(P(t)=\quad \frac{t^{4}}{4}+\frac{t^{3}}{3}\)
4. \(P(t)=\left(1+t^{2}\right)^{2}\)
5. \(P(t)=31 t^{52}-82 t^{241}+\pi t^{314}\)
6. \(P(t)=2^{5}+17 t^{5}\)
7. \(P(t)=\sqrt{2}-\frac{t^{7}}{427}+18 t^{35}\)
8. \(P(t)=17^{3}-\frac{t^{23}}{690}+5 t^{705}\)

Exercise 3.5.19 Find values of t for which \(P ^{\prime} (t) = 0\) for:

1. \(P(t)=t^{2}-10 t+35\)
2. \(P(t)=t^{3}-3 t+8\)
3. \(P(t)=5 t^{2}-t+1\)
4. \(P(t)=t^{3}-6 t^{2}+9 t+7\)
5. \(P(t)=7 t^{4}-56 t^{2}+8\)
6. \(P(t)=t+\frac{1}{t} \quad t>0\)
7. \(P(t)=\frac{t}{2}+\frac{2}{t}\)
8. \(P(t)=\frac{t^{3}}{3}-t^{2}+t\)

Exercise 3.5.20 Suppose that in the problem of Example 3.5.6, the work of building a web is proportional to d 4 , the fourth power of the diameter, d. Then the energy available to the spider is \[E=k_{1} d^{2}-k_{2} d^{4}\] Assume that \(k_{1} = 0.01\) and \(k_2 = 0.000001\).

1. Draw a graph of \(E = 0.01 d^{2} − 0.000001 d^{4}\) for \(−10 \leq d \leq 110\).
2. Find \(E \prime (d)\) for \(E(d) = 0.01 d^{2} − 0.000001 d^{4}\).
3. Find two numbers, \(d\), for which \(E ^{\prime} (d) = 0\).
4. Find the highest point of the graph between \(d = 0\) and \(d = 100\).

Exercise 3.5.21 Consider a territorial bird that harvests only in its defended territory (assumed to be circular in shape). The amount of food available can be assumed to be proportional to the area of the territory and therefore proportional to \(d^2\) , the square of the diameter of the territory. Assume that the food gathered is proportional to the amount of food available times the time spent gathering food. Let the unit of time be one day, and suppose the amount of time spent defending the territory is proportional to the length of the territory boundary and therefore equal to \(k \times d\) for some constant, \(k\). Then \(1 − k d\) is the amount of time available to gather food, and the amount, \(F\) of food gathered will be

\[F=k_{2} d^{2}(1-k d)\]

Find the value of \(d\) that will maximize the amount of food gathered.