Chapter 2

You measure the radius \(r\) of a spherical tumor to be \(1.2 \mathrm{~cm}\) with an error no greater than \(3 \%\). Use calculus to estimate the error incurred by using this approximate value of \(r\) in the formula \(S=4 \pi r^{2}\) to compute the surface area of the tumor.

One model of the cardiovascular system relates \(V(t)\), the stroke volume of blood in the aorta at a time \(t\) during systole (the contraction phase), to the pressure \(P(t)\) in the aorta during systole by the equation \(V(t)=\left[C_{1}+C_{2} P(t)\right]\left(\frac{3 t^{2}}{T^{2}}-\frac{2 t^{3}}{T^{3}}\right)\) where \(C_{1}\) and \(C_{2}\) are positive constants and \(T\) is the (fixed) time length of the systole phase. Find a relationship between the rates \(\frac{d V}{d t}\) and \(\frac{d P}{d t}\).

When the price of a certain commodity is \(p\) dollars per unit, consumers demand \(x\) hundred units of the commodity, where $$ 75 x^{2}+17 p^{2}=5,300 $$ How fast is the demand \(x\) changing with respect to time when the price is \( 7\) and is decreasing at the rate of 75 cents per months? (That is, \(\frac{d p}{d t}=-0.75\).)

At noon, a truck is at the intersection of two roads and is moving north at \(70 \mathrm{~km} / \mathrm{hr}\). An hour later, a car passes through the same intersection, traveling east at \(105 \mathrm{~km} / \mathrm{hr}\). How fast is the distance between the car and truck changing at 2 P.M.?

A car is traveling at \(88 \mathrm{ft} / \mathrm{sec}\) when the driver applies the brakes to avoid hitting a child. After \(t\) seconds, the car is \(s=88 t-8 t^{2}\) feet from the point where the brakes were applied. How long does it take for the car to come to a stop, and how far does it travel before stopping?

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at \(8: 00\) A.M. will have produced \(Q(t)=-t^{3}+9 t^{2}+12 t\) units \(t\) hours later. a. Compute the worker's rate of production \(R(t)=Q^{\prime}(t)\). b. At what rate is the worker's rate of production changing with respect to time at 9:00 A.M.? c. Use calculus to estimate the change in the worker's rate of production between \(9: 00\) and 9:06 A.M. d. Compute the actual change in the worker's rate of production between \(9: 00\) and 9:06 A.M.

Sand is leaking from a bag in such a way that after \(t\) seconds, there are $$ S(t)=50\left(1-\frac{t^{2}}{15}\right)^{3} $$ pounds of sand left in the bag. a. How much sand was originally in the bag? b. At what rate is sand leaking from the bag after 1 second? c. How long does it take for all the sand to leak from the bag? At what rate is the sand leaking from the bag at the time it empties?

It is projected that \(t\) months from now, the average price per unit for goods in a certain sector of the economy will be \(P\) dollars, where $$ P(t)=-t^{3}+7 t^{2}+200 t+300 $$ a. At what rate will the price per unit be increasing with respect to time 5 months from now? b. At what rate will the rate of price increase be changing with respect to time 5 months from now? c. Use calculus to estimate the change in the rate of price increase during the first half of the sixth month. d. Compute the actual change in the rate of price increase during the first half of the sixth month.

At a certain factory, approximately \(q(t)=t^{2}+50 t\) units are manufac- tured during the first \(t\) hours of a production run, and the total cost of manufacturing \(q\) units is \(C(q)=0.1 q^{2}+10 q+400\) dollars. Find the rate at which the manufacturing cost is changing with respect to time 2 hours after production commences.

It is estimated that the monthly cost of producing \(x\) units of a particular commodity is \(C(x)=0.06 x+3 x^{1 / 2}+20\) hundred dollars. Suppose production is decreasing at the rate of 11 units per month when the monthly production is 2,500 units. At what rate is the cost changing at this level of production?

Estimate the largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula \(S=4 \pi r^{2}\) to be no greater than \(8 \%\).

A soccer ball made of leather \(1 / 8\) inch thick has an inner diameter of \(8.5\) inches. Estimate the volume of its leather shell.

A car traveling north at \(60 \mathrm{mph}\) and a truck traveling east at \(45 \mathrm{mph}\) leave an intersection at the same time. At what rate is the distance between them changing 2 hours later?

A child is flying a kite at a height of 80 feet above her hand. If the kite moves horizontally at a constant speed of \(5 \mathrm{ft} / \mathrm{sec}\), at what rate is string being paid out when the string is 100 feet long?

A person stands at the end of a pier 8 feet above the water and pulls in a rope attached to a buoy. If the rope is hauled in at the rate of \(2 \mathrm{ft} / \mathrm{min}\), how fast is the buoy moving in the water when it is 6 feet from the pier?

A 10 -foot-long ladder leans against the side of a wall. The top of the ladder is sliding down the wall at the rate of \(3 \mathrm{ft} / \mathrm{sec}\). How fast is the foot of the ladder moving away from the building when the top is 6 feet above the ground?

A lantern falls from the top of a building in such a way that after \(t\) seconds, it is \(h(t)=150-16 t^{2}\) feet above ground. A woman 5 feet tall originally standing directly under the lantern sees it start to fall and walks away at the constant rate of \(5 \mathrm{ft} / \mathrm{sec}\). How fast is the length of the woman's shadow changing when the lantern is 10 feet above the ground?

A baseball diamond is a square, 90 feet on a side. A runner runs from second base to third at \(20 \mathrm{ft} / \mathrm{sec}\). How fast is the distance \(s\) between the runner and home plate changing when he is 15 feet from third base?

Suppose the total manufacturing cost \(C\) at a certain factory is a function of the number \(q\) of units produced, which in turn is a function of the number \(t\) of hours during which the factory has been operating. a. What quantity is represented by the derivative \(\frac{d C}{d q} ?\) In what units is this quantity measured? b. What quantity is represented by the derivative \(\frac{d q}{d t} ?\) In what units is this quantity measured? c. What quantity is represented by the product \(\frac{d C}{d q} \frac{d q}{d t} ?\) In what units is this quantity measured?

Find all the points \((x, y)\) on the graph of the function \(y=4 x^{2}\) with the property that the tangent to the graph at \((x, y)\) passes through the point \((2,0)\).

Suppose \(y\) is a linear function of \(x\); that is, \(y=m x+b\). What will happen to the percentage rate of change of \(y\) with respect to \(x\) as \(x\) increases without bound? Explain.

Find an equation for the tangent line to the curve $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $$ at the point \(\left(x_{0}, y_{0}\right)\).

Let \(f(x)=(3 x+5)\left(2 x^{3}-5 x+4\right)\). Use a graphing utility to graph \(f(x)\) and \(f^{\prime}(x)\) on the same set of coordinate axes. Use TRACE and \(\mathbf{Z O O M}\) to find where \(f^{\prime}(x)=0\).

Use a graphing utility to graph \(f(x)=\frac{2 x+3}{1-3 x}\) and \(f^{\prime}(x)\) on the same set of coordinate axes. Use TRACE and ZOOM to find where \(f^{\prime}(x)=0\).

The curve \(y^{2}(2-x)=x^{3}\) is called a cissoid. a. Use a graphing utility to sketch the curve. b. Find an equation for the tangent line to the curve at all points where \(x=1\). c. What happens to the curve as \(x\) approaches 2 from the left? d. Does the curve have a tangent line at the origin? If so, what is its equation?

An object moves along a straight line in such a way that its position at time \(t\) is given by \(s(t)=t^{5 / 2}\left(0.73 t^{2}-3.1 t+2.7\right)\) for \(0 \leq t \leq 2\) a. Find the velocity \(v(t)\) and the acceleration \(a(t)\), and then use a graphing utility to graph \(s(t), v(t)\), and \(a(t)\) on the same axes for \(0 \leq t \leq 2\). b. Use your calculator to find a time when \(v(t)=0\) for \(0 \leq t \leq 2\). What is the object's position at this time? c. When does the smallest value of \(a(t)\) occur? Where is the object at this time and what is its velocity?