Chapter 3

Use Definition of the Derivative 3.22, \(\lim _{b \rightarrow t} \frac{P(b)-P(t)}{b-t},\) to compute the rates of change of the following functions, \(P\). a. \(\quad P(t)=t^{3}\) b. \(P(t)=5 t^{2}\) c. \(P(t)=\frac{t^{3}}{4}\) d. \(\quad P(t)=t^{2}+t^{3}\) e. \(\quad P(t)=2 \sqrt{t}\) f. \(P(t)=\sqrt{2 t}\) g. \(\quad P(t)=7\) h. \(P(t)=5-2 t\) i. \(\quad P(t)=\frac{1}{1+t}\) j. \(\quad P(t)=\frac{1}{3 t}\) k. \(\quad P(t)=5 t^{7}\) l. \(\quad P(t)=\frac{1}{t^{2}}\) m. \(P(t)=\frac{1}{(3 t+1)^{2}}\) n. \(P(t)=\sqrt{t+3}\) o. \(P(t)=\frac{1}{\sqrt{t+1}}\)

Compute \(P^{\prime}, P^{\prime \prime}\) and \(P^{\prime \prime \prime}\) for the following functions. a. \(P(t)=17\) b. \(P(t)=t\) c. \(P(t)=t^{2}\) d. \(P(t)=t^{3}\) e. \(\quad P(t)=t^{1 / 2}\) f. \(P(t)=t^{-1}\) g. \(P(t)=t^{8}\) h. \(P(t)=t^{125}\) i. \(P(t)=t^{5 / 2}\)

Use Equation 3.22 , $$F^{\prime}(x)=\lim _{b \rightarrow x} \frac{F(b)-F(x)}{b-x},$$ to compute \(F^{\prime}(x)\) for a. \(\quad F(x)=x^{2}\) b. \(\quad F(x)=2 x^{2}\) c. \(\quad F(x)=x^{2}+1\) d. \(\quad F(x)=x^{3}\) e. \(\quad F(x)=4 x^{3}\) f. \(\quad F(x)=x^{3}-1\) g. \(\quad F(x)=x^{2}+x\) h. \(\quad F(x)=x^{2}+x^{3}\) i. \(\quad F(x)=3 x+1\) j. \(\quad F(x)=\sqrt{x}\) k. \(\quad F(x)=4 \sqrt{x}\) l. \(\quad F(x)=4+\sqrt{x}\) m. \(\quad F(x)=5\) n. \(\quad F(x)=\frac{1}{x}\) o. \(\quad F(x)=5+\frac{1}{x}\) p. \(\quad F(x)=\frac{1}{x^{2}}\) q. \(\quad F(x)=\frac{5}{x^{2}}\) r. \(\quad F(x)=5+\frac{1}{x^{2}}\)

a. Find a number, \(\delta>0,\) so that if \(x\) is a number and \(0<|x-2|<\delta\) then \(|2 x-4|<0.01\). b. Find a number, \(\delta\), so that if \(x\) is a number and \(0<|x-2|<\delta\) then \(\left|x^{2}-4\right|<0.01\). c. Find a number, \(\delta\), so that if \(x\) is a number and \(0<|x-2|<\delta\) then \(\left|\frac{1}{x}-\frac{1}{2}\right|<0.01\). d. Find a number, \(\delta,\) so that if \(x\) is a number and \(0<|x-2|<\delta\) then \(\left|x^{3}-8\right|<0.01\). e. Find a number, \(\delta\), so that if \(x\) is a number and \(0<|x-2|<\delta\) then \(\left|\frac{x}{x+1}-\frac{2}{3}\right|<0.01\). f. Find a number, \(\delta>0\), so that if \(x\) is a number and \(0<|x-9|<\delta\) then \(|\sqrt{x}-3|<0.01\). g. Find a number, \(\delta\), so that if \(x\) is a number and \(0<|x-8|<\delta\) then \(|\sqrt[3]{x}-2|<0.01\). h. Find a number, \(\delta>0\), so that if \(x\) is a number and \(0<|x-2|<\delta\) then \(\left|x^{4}-3 x-13\right|<0.01\)

Let functions \(D, E, F, G,\) and \(H\) be defined by $$\begin{aligned}&D(x)=|x|\\\&\text { for all } x\\\&E(x)=\left\\{\begin{aligned}-1 & \text { for } x<0 \\\0 & \text { for } x=0 \\\1 & \text { for } 0

Find the acceleration of a particle at time \(t\) whose position, \(P(t),\) on an axis is described by a. \(P(t)=15\) b. \(\quad P(t)=5 t+7\) c. \(P(t)=-4.9 t^{2}+22 t+5\) d. \(P(t)=t-\frac{t^{3}}{6}+\frac{t^{5}}{120}\)

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

Use the equation, $$F^{\prime}(x)=\lim _{h \rightarrow 0} \frac{F(x+h)-F(x)}{h}$$ to comupute \(F^{\prime}(x)\) for: a. \(\quad F(x)=x^{2}\) b. \(\quad F(x)=3 x^{2}\) c. \(\quad F(x)=x^{2}+5\) d. \(\quad F(x)=x^{-1}\) e. \(\quad F(x)=2 x^{-1}\) f. \(\quad F(x)=x^{-1}-7\)

\(\begin{array}{ll}& \text { a. Find a number, } \delta>0, \text { so that }\end{array}\) if \(x\) is a number and \(\quad|x-3|<\delta \quad\) then \(\left|\frac{x^{2}-9}{x-3}-6\right|<0.01\) b. Find a number, \(\delta>0\), so that if \(x\) is a number and \(\quad|x-4|<\delta \quad\) then \(\left|\frac{\sqrt{x}-2}{x-4}-\frac{1}{4}\right|<0.01\) c. Find a number, \(\delta>0\), so that if \(x\) is a number and \(\quad|x-2|<\delta\) then \(\left|\frac{\frac{1}{x}-\frac{1}{2}}{x-2}+\frac{1}{4}\right|<0.01\) d. Find a number, \(\delta>0\), so that if \(x\) is a number and \(\quad|x-1|<\delta\) then \(\left|\frac{x^{2}+x-2}{x-1}-3\right|<0.01\)

Compute \(P^{\prime}, P^{\prime \prime}\) and \(P^{\prime \prime \prime}\) and \(P^{(4)}\) for \(P(t)=a+b t+c t^{2}+d t^{3}\).

a. Use Equations 3.11 and 3.15 , $$\lim _{x \rightarrow a} x=a \quad \text { and } \quad \lim _{x \rightarrow a} F_{1}(x) \times F_{2}(x)=\left(\lim _{x \rightarrow a} F_{1}(x)\right) \times\left(\lim _{x \rightarrow a} F_{2}(x)\right),$$ to show that $$\lim _{x \rightarrow a} x^{2}=a^{2}$$ b. Show that $$\lim _{x \rightarrow a} x^{3}=a^{3}$$ c. Show by induction that if \(n\) is a positive integer, $$\lim _{x \rightarrow a} x^{n}=a^{n}$$

Find an equation of the tangent to the graph of \(P\) at the indicated points. Draw the graph \(P\) and the tangent. a. \(P(t)=t^{4} \quad\) at \(\quad(1,1)\) b. \(P(t)=t^{12}\) at \(\quad(1,1)\) c. \(P(t)=t^{1 / 2} \quad\) at \(\quad(4,2)\) d. \(P(t)=\frac{5}{2}\) at e. \(P(t)=\sqrt{1+t}\) at (8,3) f. \(P(t)=\frac{1}{2 t}\) at \(\quad\left(\frac{1}{2}, 1\right)\)

Suppose \(m\) is a positive integer and \(u(t)=t^{-m}=1 / t^{m}\) for \(t \neq 0 .\) Show that \(u^{\prime}(t)=-m t^{-m-1}\), thus proving the \(t^{n}\) rule for negative integers. Begin your argument with $$\begin{aligned}u^{\prime}(t) &=\lim _{b \rightarrow t} \frac{\frac{1}{b^{m}}-\frac{1}{t^{m}}}{b-t} \\ &=\lim _{b \rightarrow t} \frac{t^{m}-b^{m}}{b^{m} t^{m}} \times \frac{1}{b-t}\end{aligned}$$

Data from Purdue University \(^{7}\) for the decrease of the titration marker phenolphthalein (Hln) in the presence of excess base are shown in Table 3.4.5. The data show the concentration of phenolphthalein that was initially at \(0.005 \mathrm{M}\) in a solution with \(0.61 \mathrm{M} \mathrm{OH}^{-}\) ion. a. Graph the data. b. Estimate the rate of change of the concentration of phenolphthalein (Hln) for each of the times shown. c. Draw a graph of the rate of reaction versus concentration of phenolphthalein. $$\begin{array}{|l|rrrrrrrr|}\hline \text { Time (sec) } & 0 & 10.5 & 22.3 & 35.7 & 51.1 & 69.3 & 91.6 & 120.4 & 160.9 \\ \text { Hln (M) } & 0.005 & 0.0045 & 0.0040 & 0.0035 & 0.0030 & 0.0025 & 0.0020 & 0.0015 & 0.0010 \\ \hline\end{array}$$

Evaluate the limits. a. \(\lim _{x \rightarrow 5} 3 x^{2}-15 x\) b. \(\lim _{x \rightarrow 0} 3 x\) c. \(\lim _{x \rightarrow 50} \pi\) d. \(\lim _{x \rightarrow-2} 3 x^{2}-15 x\) e. \(\lim _{x \rightarrow-\pi} x\) f. \(\lim _{x \rightarrow \pi} 50\) g. \(\lim _{x \rightarrow-5} \frac{x-1}{x-1}\) h. \(\lim _{x \rightarrow 1} \frac{x^{2}-2 x+1}{x-1}\) i. \(\quad \lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}\) j. \(\quad \lim _{x \rightarrow 4} \frac{x}{\sqrt{x+2 x^{2}}}\) k. \(\lim _{x \rightarrow 1} \sqrt{\frac{x-2}{x^{3}+1}}\) 1\. \(\lim _{x \rightarrow 1} \sqrt{\frac{x-3}{x^{3}-27}}\)

Sketch the graph of \(y=\sqrt[3]{x}\) for \(-1 \leq x \leq 1\). Does the graph have a tangent at (0,0) ? Remember, Your vote counts.

If \(b\) approaches 3 a. \(b\) approaches _________. b. \(2 \div b\) approaches _________. c. \(\pi\) approaches _________. d. \(\frac{2}{\sqrt{b}+\sqrt{3}}\) approaches _________. e. \(b^{3}+b^{2}+b\) approaches _________. f. \(\frac{b}{1+b}\) approaches _________. g. \(2^{b}\) approaches _________. h. \(\log _{3} b\) approaches _________.

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?

Show that for \(s(t)=\frac{g}{2} t^{2}+v_{0} t+\gamma, s^{\prime}(t)=g t+v_{0}\).

The square function, \(S(t)=t^{2}, t \geq 0,\) and the square root function, \(R(t)=\sqrt{t}, t \geq 0,\) are each inverses of the other. See Figure Ex. 3.3.8. Compare the slopes of the tangents to \(S\) at the points \((2,4),(3,9),\) and (4,16) with the slopes of \(R\) at the respectively corresponding points, \((4,2),(9,3),\) and (16,4) of \(R\). Compare the slope of the graph of \(S\) at the point \(\left(a, a^{2}\right), a>0\) with the slope of the graph of \(R\) at the corresponding point \(\left(a^{2}, a\right)\).

Evaluate \(\gamma\) if a. \(s(t)=5 t^{2}+\gamma \quad\) and \(\quad s(0)=15\) b. \(s(t)=-8 t^{2}+12 t+\gamma\) and \(\quad s(0)=11\) c. \(s(t)=-8 t^{2}+12 t+\gamma \quad\) and \(\quad s(1)=11\) d. \(s(t)=-\frac{849}{2} t^{2}+126 t+\gamma\) and \(s(0.232)=245.9\)

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?

Show that a. \(P(t)=5 t+3\) satisfies \(P(0)=3\) and \(P^{\prime}(t)=5\) b. \(\quad P(t)=8 t+2\) satisfies \(P(0)=2\) and \(P^{\prime}(t)=8\) c. \(P(t)=t^{2}+3 t+7 \quad\) satisfies \(P(0)=7\) and \(P^{\prime}(t)=2 t+3\) d. \(P(t)=-2 t^{2}+5 t+8\) satisfies \(P(0)=8\) and \(P^{\prime}(t)=-4 t+5\) e. \(\quad P(t)=(3 t+4)^{2}\) satisfies \(P(0)=16\) and \(P^{\prime}(t)=6 \sqrt{P(t)}\) f. \(P(t)=(5 t+1)^{2}\) satisfies \(P(0)=1\) and \(P^{\prime}(t)=10 \sqrt{P(t)}\) g. \(\quad P(t)=(1-2 t)^{-1}\) satisfies \(P(0)=1\) and \(P^{\prime}(t)=2(P(t))^{2}\) h. \(P(t)=\frac{5}{1-15 t}\) satisfies \(P(0)=5\) and \(P^{\prime}(t)=3(P(t))^{2}\) i. \(P(t)=(6 t+9)^{1 / 2} \quad\) satisfies \(P(0)=3\) and \(P^{\prime}(t)=3 / P(t)\) j. \(P(t)=(4 t+4)^{1 / 2} \quad\) satisfies \(P(0)=2\) and \(P^{\prime}(t)=2 / P(t)\) k. \(P(t)=(4 t+4)^{3 / 2} \quad\) satisfies \(P(0)=8 \quad\) and \(\quad P^{\prime}(t)=6 \sqrt[3]{P(t)}\) For parts \(\mathrm{g}-\mathrm{k}\), use the Definition of Derivative 3.2 .2 to compute \(P^{\prime}\).

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?

\(\lim _{x \rightarrow a} \frac{F(x)-F(a)}{x-a}\) a. \(\quad F(x)=x^{2} \quad a=-2\) b. \(\quad F(x)=17 \quad a=0\) c. \(\quad F(x)=2 x^{3} \quad a=2\) d. \(\quad F(x)=x^{2}+2 x \quad a=1\) e. \(\quad F(x)=\frac{1}{x} \quad a=\frac{1}{2}\) f. \(\quad F(x)=3 x^{2}-5 x \quad a=7\) g. \(\quad F(x)=3 \sqrt{x} \quad a=4\) h. \(\quad F(x)=x^{2}+2 x+1 \quad a=-1\) i. \(F(x)=\frac{4}{x}+5 \quad a=2 \quad\) j. \(\quad F(x)=x^{6} \quad a=2\) k. \(\quad F(x)=\frac{1}{x^{3}} \quad a=2\) l. \(F(x)=x^{10}\) \(a=2\) m. \(\quad F(x)=\frac{4}{x^{5}} \quad a=2\) n. \(\quad F(x)=x^{67} \quad a=1\)

Technology. Suppose plasma penicillin concentration in a patient following injection of 1 gram of penicillin is observed to be $$P(t)=2002^{-0.03 t}$$ where \(t\) is time in minutes and \(P(t)\) is \(\mu \mathrm{g} / \mathrm{ml}\) of penicillin. Use the following steps to approximate the rate at which the penicillin level is changing at time \(t=5\) minutes and at \(t=0\) minutes. a. \(\mathrm{t}=5\) minutes. Draw the graph of \(P(t)\) vs \(t\) for \(4.9 \leq t \leq 5.1 .\) (The graph should appear to be a straight line on this short interval.) b. Complete the table on the left, computing the average rates of change of penicillin level. $$\begin{array}{l|l}b & \frac{P(b)-P(5)}{b-5} \\\\\hline 4.9 & -3.7521 \\\4.95 & \\\4.99 & \\\4.995 & \\\& \\\5.005 & \\\5.01 & \\\5.05 & \\\5.1 & -3.744\end{array}$$ $$\begin{array}{l|l}b & \frac{P(b)-P(0)}{b-0} \\\\\hline-0.1 & \text { OMIT } \\\\-0.05 & \text { OMIT } \\ -0.01 & \text { OMIT } \\\\-0.005 & \text { OMIT } \\\0.005 & \\\0.01 & \\\0.05 & \\\0.1 & -4.155\end{array}$$ c. What is your best estimate of the rate of change of penicillin level at the time \(t=5\) minutes? Include units in your answer. d. \(\mathbf{t}=\mathbf{0}\) minutes. Complete the second table above. The OMIT entries in the second table refer to the fact that the level of penicillin, \(P(t),\) may not be given by the formula for negative values of time, \(t .\) What is your best estimate of the rate of change of penicillin level at the time \(t=0\) minutes?

Add the equations, $$\begin{aligned} H_{2}-H_{1} &=\frac{-849}{2}\left(t_{2}^{2}-t_{1}^{2}\right)+126\left(t_{2}-t_{1}\right) \\\ H_{3}-H_{2} &=\frac{-849}{2}\left(t_{3}^{2}-t_{2}^{2}\right)+126\left(t_{3}-t_{2}\right) \end{aligned}$$ $$H_{n}-H_{n-1}=\frac{-849}{2}\left(t_{n}^{2}-t_{n-1}^{2}\right)+126\left(t_{n}-t_{n-1}\right) $$to obtain $$H_{n}-H_{1}=\frac{-849}{2}\left(t_{n}^{2}-t_{1}^{2}\right)+126\left(t_{n}-t_{1}\right)$$

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?

In a chemical reaction of the form$$2 \mathrm{~A}+\mathrm{B} \longrightarrow \mathrm{A}_{2} \mathrm{~B} $$, where the reaction does not involve intermediate compounds, the reaction rate is proportional to [A] \(^{2}\) [B] where [A] and [B] denote, respectively, the concentrations of the components A and B. Let \(a\) and \(b\) denote \([\mathrm{A}]\) and \([\mathrm{B}]\), respectively, and assume that \([\mathrm{B}]\) is much greater than \([\mathrm{A}]\) so that \((b(t) \gg a(t) .\) The rate at which \(a\) changes may be written $$a^{\prime}(t)=-k(a(t))^{2} b(t)=-K(a(t))^{2}$$ We have assumed that \(b(t)\) is (almost) constant because [A] is the limiting concentration of the reaction. Let $$a(t)=\frac{a_{0}}{1-a_{0} K t}$$ Show that \(\quad a(0)=a_{0}\). Use the Definition of Derivative 3.22 , $$a^{\prime}(t)=\lim _{b \rightarrow t} \frac{a(b)-a(t)}{b-t}$$ to compute \(a^{\prime}(t)\). Then compute \((a(t))^{2}\) and show that $$a^{\prime}(t)=-K(a(t))^{2}$$

Show that the derivatives of cubic functions are quadratic functions.

Show that for any quadratic function, \(Q(t)=a+b t+c t^{2}(a, b\) and \(c\) are constants), and any interval, \([u, v]\), the average rate of change of \(Q\) on \([u, v]\) is equal to the rate of change of \(Q\) at the midpoint, \((u+v) / 2,\) of \([u, v]\).

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 \(/ \mathrm{sec})\), in which case the height of the ball in feet will be \(s(t)=-16 t^{2}+88 t\) 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?

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 \(s(t)=10-4.9 t^{2}\) meters above the ground. How fast is the squirrel falling when it hits the ground?

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 \(10 \mathrm{~cm}\) 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

Let \(P(t)=-3+5 t-2 t^{2}\). Cite the formulas that justify steps \((i)-(v i)\) below: $$\begin{aligned} P^{\prime}(t) &=\left[-3+5 t+(-2) t^{2}\right]^{\prime} \\ &=[-3]^{\prime}+[5 t]^{\prime}+\left[(-2) t^{2}\right]^{\prime} \\ &=0+[5 \times t]^{\prime}+\left[(-2) t^{2}\right]^{\prime} \\ &=0+5[t]^{\prime}+(-2)\left[t^{2}\right]^{\prime} \\ &=0+5 \times 1+(-2)\left[t^{2}\right]^{\prime} \\ &=0+5 \times 1+(-2) \times 2 t \\ &=5-4 t\end{aligned}$$

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. a. \(P(t)=15 t^{2}-32 t^{6}\) c. \(P(t)=\frac{t^{4}}{4}+\frac{t^{3}}{3}\) b. \(P(t)=1+t+t^{2}+t^{3}\) d. \(P(t)=\left(1+t^{2}\right)^{2}\) e. \(P(t)=31 t^{52}-82 t^{241}+\pi t^{314}\) f. \(P(t)=2^{5}+17 t^{5}\) g. \(P(t)=\sqrt{2}-\frac{t^{7}}{427}+18 t^{35}\) h. \(P(t)=17^{3}-\frac{t^{23}}{690}+5 t^{705}\)

Find values of \(t\) for which \(P^{\prime}(t)=0\) for: a. \(P(t)=t^{2}-10 t+35\) c. \(P(t)=5 t^{2}-t+1\) b. \(P(t)=t^{3}-3 t+8\) d. \(P(t)=t^{3}-6 t^{2}+9 t+7\) e. \(P(t)=7 t^{4}-56 t^{2}+8\) f. \(P(t)=t+\frac{1}{t} \quad t>0\) g. \(P(t)=\frac{t}{2}+\frac{2}{t}\) h. \(P(t)=\quad \frac{t^{3}}{3}-t^{2}+t\)

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

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.