Chapter 4

Compute the derivative of \(P\). a. \(P(t)=3 t^{2}-2 t+7\) b. \(P(t)=t+\frac{2}{t}\) c. \(\quad P(t)=\sqrt{t+2}\) d. \(P(t)=\left(t^{2}+1\right)^{5}\) e. \(\quad P(t)=\sqrt{2 t+1}\) f. \(\quad P(t)=2 t^{-3}-3 t^{-2}\) h. \(P(t)=(1+\sqrt{t})^{-1}\) g. \(P(t)=\frac{5}{t+5}\) i. \(P(t)=(1+2 t)^{5}\) j. \(P(t)=(1+3 t)^{1 / 3}\) k. \(P(t)=\frac{1}{1+\sqrt{t}}\) l. \(P(t)=\sqrt[3]{1+2 t}\)

For those points that are on the graph, find the slopes of the tangents to the graph of a. \(\quad \frac{x^{2}}{18}+\frac{y^{2}}{8}=1\) at the points (3,2) and (-3,2) b. \(\frac{2 x^{2}}{35}+\frac{3 y^{2}}{35}=1\) at the points \((4,1), \quad(-3,-2),\) and (4,-1)

Compute \(y^{\prime}(x)\) for a. \(\quad y=2 x^{3}-5\) b. \(y=\frac{2}{x^{2}}\) c. \(y=\frac{5}{(x+1)^{2}}\) d. \(y=\left(1+x^{2}\right)^{0.5}\) e. \(\quad y=\sqrt{1-x^{2}}\) f. \(y=\left(1-x^{2}\right)^{-0.5}\) g. \(\quad y=(2-x)^{4}\) h. \(y=\left(3-x^{2}\right)^{4}\) i. \(y=\frac{1}{\sqrt{7-x^{2}}}\) j. \(y=\left(1+(x-2)^{2}\right)^{2}\) k. \(y=(1+3 x)^{1.5} \quad\) l. \(\quad y=\frac{1}{\sqrt{16-x^{2}}}\) \(\mathrm{m} . \quad y=\sqrt{9-(x-4)^{2}}\) n. \(y=\left(9-x^{2}\right)^{1.5}\) o. \(y=\sqrt[3]{1-3 x}\)

Compute \(P^{\prime}(t)\) for a. \(\quad P(t)=\left(2+t^{2}\right)^{4}\) b. \(\quad P(t)=(1+\sin t)^{3}\) c. \(\quad P(t)=\left(t^{4}+5\right)^{2}\) d. \(\quad P(t)=\left(6 t^{7}+5^{4}\right)^{9}\) e. \(P(t)=\left(\frac{t}{8}+t^{2}\right)^{2}\) f. \(\quad P(t)=\left(t^{2}+\sin t\right)^{13}\) g. \(\quad P(t)=\left(\frac{1}{t}+t\right)^{2}\) h. \(P(t)=\left(\frac{5}{t}+\frac{t}{3}\right)^{2}\) i. \(P(t)=\frac{1}{t+5}\) j. \(P(t)=\frac{2}{(1+t)^{2}}\)

In "Natural History", March, \(1996,\) Neil de Grass Tyson discusses the discovery of an astronomical object called a "brown dwarf". "We have suspected all along that brown dwarfs were out there. One reason for our confidence is the fundamental theorem of mathematics that allows you to declare that if you were once \(3^{\prime} 8^{\prime \prime}\) tall and are now \(5^{\prime} 8 "\) tall, then there was a moment when you were \(4^{\prime} 8^{\prime \prime}\) tall (or any other height in between). An extension of this notion to the physical universe allows us to suggest that if round things come in low-mass versions (such as planets) and high-mass versions (such as stars) then there ought to be orbs at all masses in between provided a similar physical mechanism made both. What fundamental theorem of mathematics is being referenced in the article about the astronomical objects called brown dwarfs? What implicit assumption is being made about the sizes of astronomical objects? (For future consideration: Is the number of 'orbs' countable?)

Find the slope of the graph of \(\sqrt{x}-\sqrt{5-y^{2}}=5\) at the point (36,-2) . Is there a slope to the graph at the point (46,1)\(?\) Figure 4.7: Solid curve: Graph of \(\sqrt{x}+\sqrt{5-y^{2}}=5\) and the points (9,1) and (4,2) for Example 4.6 .1 . Dashed curve: Graph of \(\sqrt{x}-\sqrt{5-y^{2}}=5\) and the point (36,-2) for Exercise 4.6 .2 .

For \(f(x)=1 / x\) a. How close must \(x\) be to 0.5 in order that \(f(x)\) is within 0.01 of \(2 ?\) b. How close must \(x\) be to 3 in order to insure that \(\frac{1}{x}\) be within 0.01 of \(\frac{1}{3} ?\) c. How close must \(x\) be to 0.01 in order to insure that \(\frac{1}{x}\) be within 0.1 of \(100 ?\)

In a square field with sides of length 1000 feet that are already fenced a farmer wants to fence two rectangular pens of equal area using 400 feet of new fence and the existing fence around the field. What dimensions of lots will maximize the area of the two pens?

If \(x\) pounds per acre of nitrogen fertilizer are spread on a corn field, the yield is $$ 200-\frac{4000}{x+25} $$ bushels per acre. Corn is worth $$\$ 6.50$$ per bushel and nitrogen costs $$\$ 0.63$$ per pound. All other costs of growing and harvesting the crop amount to $$\$$ 760$ per acre, and are independent of the amount of nitrogen fertilizer applied. How much nitrogen per acre should be used to maximize the net dollar return per acre? Note: The parameters of this problem are difficult to keep up to date.

Shown Figure Ex. 4.4 .3 is the ellipse, $$ \frac{2 x^{2}}{35}+\frac{3 y^{2}}{35}=1 $$ and tangents to the ellipse at (2,3) and at (4,-1) . a. Find the slopes of the tangents. b. Find equations of the tangents. c. Find the point of intersection of the tangents. Figure for Exercise 4.4 .3 Graph of the ellipse \(2 x^{2} / 35+3 y^{2} / 35=1\) and tangents to the ellipse at the points (2,3) and (4,-1) .

For for the function \(P(t)=|t|\) for all \(t\) compute \(P^{\prime-}(0)\) and \(P^{\prime+}(0)\).

You must cross a river that is 50 meters wide and reach a point on the opposite bank that is \(1 \mathrm{~km}\) up stream. You can travel \(6 \mathrm{~km}\) per hour along the river bank and \(1 \mathrm{~km}\) per hour in the river. Describe a path that will minimize the amount of time required for your trip. Neglect the flow of water in the river.

Draw the graph and find the slopes of the tangents to the graph of a. \(x^{2}-2 y^{2}=1 \quad\) at the points (3,2) and (-3,-2) b. \(2 x^{4}+3 y^{4}=35 \quad\) at the points \((2,1), \quad(1,-2),\) and (2,-1) c. \(\sqrt{|x|}+\sqrt{|y|}=5 \quad\) at the points (9,4) and (1,-16) d. \(\quad \sqrt{x}+\sqrt[3]{y}=9\) at the points \((64,1), \quad(36,27), \quad\) and \(\quad(16,125)\) e. \(x^{2}+y^{2}=(x+y)^{2} \quad\) at the points \(\quad(1,0)\) and (0,1) f. \(\quad\left(x^{2}+4\right) y=24 \quad\) at the points (2,3) and (0,6) g. \(x^{3}+y^{3}=(x+y)^{3}\) at the points (1,-1) and (-2,0) h. \(x^{4}+x^{2} y^{2}=20 y^{2} \quad\) at the points (2,1) and (2,-1)

Suppose \(p\) and \(q\) are integers and \(u\) is a positive function that has a derivative at all numbers \(t\). Assume that \(\left[u^{\frac{p}{q}}(t)\right]^{\prime}\) exists. Give reasons for the steps \((i)-(i v)\) below that show $$ \left[u^{\frac{p}{q}}(t)\right]^{\prime}=\frac{p}{q} u^{\frac{p}{q}-1}(t) \times u^{\prime}(t) $$ Let $$ v(t)=u^{\frac{p}{q}}(t) $$ Then $$ \begin{aligned} v^{q}(t) &=u^{p}(t) \\ \left[v^{q}(t)\right]^{\prime} &=\left[u^{p}(t)\right]^{\prime} \\ q v^{q-1}(t) \times v^{\prime}(t) &=p u^{p-1}(t) \times u^{\prime}(t) \\ v^{\prime}(t) &=\frac{p u^{p-1}(t)}{q} \frac{p}{\left(u^{\frac{p}{q}}\right)^{q-1}} \times u^{\prime}(t) \\ \left[u^{\frac{p}{q}}(t)\right]^{\prime} &=\frac{p}{q} u^{\frac{p}{q}-1}(t) \times u^{\prime}(t) \end{aligned} $$

Exercise 4.1 .5 a. Draw the graph of \(y_{1}\). b. Find a number \(A\) such that the graph of \(y_{2}\) is continuous. $$ \text { a. } \quad y_{1}(x)=\left\\{\begin{array}{rll} x^{2} & \text { for } x<2 \\ 3-x & \text { for } 2 \leq x \end{array} \quad\right. \text { b. } y_{2}(x)=\left\\{\begin{aligned} x^{2} & \text { for } x<2 \\ A-x & \text { for } 2 \leq x \end{aligned}\right. $$

Is the temperature of the water in a lake a continuous function of depth? Write a paragraph discussing water temperature as a function of depth in a lake and how knowledge of water temperature assists in the location of fish.

The function, \(f(x)=\sqrt[3]{x}\) is continuous. 1\. How close must \(x\) be to 1 in order to insure that \(f(x)\) is within 0.1 of \(f(1)=1\) (that is, to insure that \(0.9

Exercise \(4.1 .9 \quad\) a. Draw the graph of a function, \(f,\) defined on the interval [1,3] such that \(f(1)=-2\) and \(f(3)=4\) b. Does your graph intersect the X-axis? c. Draw a graph of of a function, \(f,\) defined on the interval [1,3] such that \(f(1)=-2\) and \(f(3)=4\) that does not intersect the X-axis. Be sure that its X-projection is all of [1,3] . d. Write equations to define a function, \(f,\) on the interval [1,3] such that \(f(1)=-2\) and \(f(3)=4\) and the graph of \(f\) does not intersect the X-axis. e. There is a theorem that asserts that the function you just defined must be discontinuous at some number in [1,3] . Identify such a number for your example. The preceding exercise illustrates a general property of continuous functions called the intermediate value property. Briefly it says that a continuous function defined on an interval that has both positive and negative values on the interval, must also be zero somewhere on the interval. In language of graphs, the graph of a continuous function defined on an interval that has a point below the X-axis and a point above the X-axis must intersect the X-axis. The proof of this property requires more than the familiar properties of addition, multiplication, and order of the real numbers. It requires the completion property of the real numbers, Axiom \(5.2 .1^{2}\).

Exercise 4.1 .10 A nutritionist studying plasma epinephrine (EPI) kinetics with tritium labeled epinephrine, \(\left[{ }^{3} \mathrm{H}\right] \mathrm{EPI},\) observes that after a bolus injection of \(\left[{ }^{3} \mathrm{H}\right] \mathrm{EPI}\) into plasma, the time-dependence of \(\left[{ }^{3} \mathrm{H}\right]\) EPI level is well approximated by \(L(t)=4 e^{-2 t}+3 e^{-t}\) where \(L(t)\) is the level of \(\left[{ }^{3} \mathrm{H}\right] \mathrm{EPI}\) t hours after infusion. Sketch the graph of \(L\). Observe that \(L(0)=7\) and \(L(2)=0.479268 .\) The intermediate value property asserts that at some time between 0 and 2 hours the level of \(\left[{ }^{3} \mathrm{H}\right] \mathrm{EPI}\) will be \(1.0 .\) At what time, \(t_{1},\) will \(L\left(t_{1}\right)=1.0 ?\) (Let \(A=e^{-t}\) and observe that \(\left.A^{2}=e^{-2 t} .\right)\)

For the function, \(f(x)=10-x^{2}\), find an an open interval, \((3-\delta, 3+\delta)\) so that \(f(x)>0\) for \(x\) in \((3-\delta, 3+\delta)\).

For the function, \(f(x)=\sin (x)\), find an an open interval, \((3-\delta, 3+\delta)\) so that \(f(x)>0\) for \(x\) in \((3-\delta, 3+\delta)\).