Chapter 2

Use the definition of the derivative to find \(f^{\prime}(x)\). $$ f(x)=x^{2}-3 x+1 $$

Use the definition of the derivative to find \(f^{\prime}(x)\). $$ f(x)=\frac{1}{x-2} $$

Find the derivative of the given function. $$ f(x)=6 x^{4}-7 x^{3}+2 x+\sqrt{2} $$

Find the derivative of the given function. $$ f(x)=x^{3}-\frac{1}{3 x^{5}}+2 \sqrt{x}-\frac{3}{x}+\frac{1-2 x}{x^{3}} $$

Find the derivative of the given function. $$ y=\frac{2-x^{2}}{3 x^{2}+1} $$

Find the derivative of the given function. $$ y=\left(x^{3}+2 x-7\right)\left(3+x-x^{2}\right) $$

Find the derivative of the given function. $$ f(x)=\left(5 x^{4}-3 x^{2}+2 x+1\right)^{10} $$

Find the derivative of the given function. $$ f(x)=\sqrt{x^{2}+1} $$

Find the derivative of the given function. $$ y=\left(\frac{x+1}{1-x}\right)^{2} $$

Find the derivative of the given function. $$ f(x)=(3 x+1) \sqrt{6 x+5} $$

Find the derivative of the given function. $$ f(x)=\frac{(3 x+1)^{3}}{(1-3 x)^{4}} $$

Find the derivative of the given function. $$ =\sqrt{\frac{1-2 x}{3 x+2}} $$

Find an equation for the tangent line to the graph of the given function at the specified point. $$ f(x)=x^{2}-3 x+2 ; x=1 $$

Find an equation for the tangent line to the graph of the given function at the specified point. $$ f(x)=\frac{4}{x-3} ; x=1 $$

Find an equation for the tangent line to the graph of the given function at the specified point. $$ f(x)=\frac{x}{x^{2}+1} ; x=0 $$

Find an equation for the tangent line to the graph of the given function at the specified point. $$ f(x)=\sqrt{x^{2}+5} ; x=-2 $$

In each of these cases, find the rate of change of \(f(t)\) with respect to \(t\) at the given value of \(t\). a. \(f(t)=t^{3}-4 t^{2}+5 t \sqrt{t}-5\) at \(t=4\) b. \(f(t)=\frac{2 t^{2}-5}{1-3 t}\) at \(t=-1\)

In each of these cases, find the rate of change of \(f(t)\) with respect to \(t\) at the given value of \(t\). a. \(f(t)=t^{3}\left(t^{2}-1\right)\) at \(t=0\) b. \(f(t)=\left(t^{2}-3 t+6\right)^{1 / 2}\) at \(t=1\)

In each of these cases, find the percentage rate of change of the function \(f(t)\) with respect to \(t\) at the given value of \(t\). a. \(f(t)=t^{2}-3 t+\sqrt{t}\) at \(t=4\) b. \(f(t)=\frac{t}{t-3}\) at \(t=4\)

In each of these cases, find the percentage rate of change of the function \(f(t)\) with respect to \(t\) at the given value of \(t\). a. \(f(t)=t^{2}(3-2 t)^{3}\) at \(t=1\) b. \(f(t)=\frac{1}{t+1}\) at \(t=0\)

Use the chain rule to find \(\frac{d y}{d x}\). a. \(y=5 u^{2}+u-1 ; u=3 x+1\) b. \(y=\frac{1}{u^{2}} ; u=2 x+3\)

Use the chain rule to find \(\frac{d y}{d x}\). a. \(y=(u+1)^{2} ; u=1-x\) b. \(y=\frac{1}{\sqrt{u}} ; u=2 x+1\)

Use the chain rule to find \(\frac{d y}{d x}\) for the given value of \(x\). a. \(y=u-u^{2} ; u=x-3 ;\) for \(x=0\) b. \(y=\left(\frac{u-1}{u+1}\right)^{1 / 2}, u=\sqrt{x-1} ;\) for \(x=\frac{34}{9}\)

Use the chain rule to find \(\frac{d y}{d x}\) for the given value a. \(y=u^{3}-4 u^{2}+5 u+2 ; u=x^{2}+1 ;\) for \(x=1\) b. \(y=\sqrt{u}, u=x^{2}+2 x-4 ;\) for \(x=2\)

Find the second derivative: a. \(f(x)=6 x^{5}-4 x^{3}+5 x^{2}-2 x+\frac{1}{x}\) b. \(z=\frac{2}{1+x^{2}}\) c. \(y=\left(3 x^{2}+2\right)^{4}\)

Find the second derivative: a. \(f(x)=4 x^{3}-3 x\) b. \(f(x)=2 x(x+4)^{3}\) c. \(f(x)=\frac{x-1}{(x+1)^{2}}\)

Find \(\frac{d y}{d x}\) by implicit differentiation. a. \(5 x+3 y=12\) b. \((2 x+3 y)^{5}=x+1\)

Find \(\frac{d y}{d x}\) by implicit differentiation. a. \(x^{2} y=1\) b. \(\left(1-2 x y^{3}\right)^{5}=x+4 y\)

Use implicit differentiation to find the slope of the line that is tangent to the given curve at the specified point. a. \(x y^{3}=8 ;(1,2)\) b. \(x^{2} y-2 x y^{3}+6=2 x+2 y ;(0,3)\)

Use implicit differentiation to find the slope of the line that is tangent to the given curve at the specified point. a. \(x^{2}+2 y^{3}=\frac{3}{x y} ;(1,1)\) b. \(y=\frac{x+y}{x-y} ;(6,2)\)

Use implicit differentiation to find \(\frac{d^{2} y}{d x^{2}}\) if \(4 x^{2}+y^{2}=1\).

Use implicit differentiation to find \(\frac{d^{2} y}{d x^{2}}\) if \(3 x^{2}-2 y^{2}=6\).

Suppose that a 5 -year projection of population trends suggests that \(t\) years from now, the population of a certain community will be \(P\) thousand, where $$ P(t)=-2 t^{3}+9 t^{2}+8 t+200 $$ a. At what rate will the population be growing 3 years from now? b. At what rate will the rate of population growth be changing with respect to time 3 years from now?

s(t) denotes the position of an object moving along a line. $$ s(t)=2 t^{3}-21 t^{2}+60 t-25 ; 1 \leq t \leq 6 $$

s(t) denotes the position of an object moving along a line. $$ s(t)=\frac{2 t+1}{t^{2}+12} ; 0 \leq t \leq 4 $$

After \(x\) weeks, the number of people using a new rapid transit system was approximately \(N(x)=6 x^{3}+500 x+8,000\). a. At what rate was the use of the system changing with respect to time after 8 weeks? b. By how much did the use of the system change during the eighth week?

It is estimated that the weekly output at a certain plant is \(Q(x)=50 x^{2}+9,000 x\) units, where \(x\) is the number of workers employed at the plant. Currently there are 30 workers employed at the plant. a. Use calculus to estimate the change in the weekly output that will result from the addition of 1 worker to the force. b. Compute the actual change in output that will result from the addition of 1 worker.

It is projected that \(t\) months from now, the population of a certain town will be \(P(t)=3 t+5 t^{3 / 2}+6,000 .\) At what percentage rate will the population be changing with respect to time 4 months from now?

At a certain factory, the daily output is \(Q(L)=20,000 L^{1 / 2}\) units, where \(L\) denotes the size of the labor force measured in worker-hours. Currently 900 worker-hours of labor are used each day. Use calculus to estimate the effect on output that will be produced if the labor force is cut to 885 worker-hours.

The gross domestic product of a certain country was \(N(t)=t^{2}+6 t+300\) billion dollars \(t\) years after 2004\. Use calculus to predict the percentage change in the GDP during the second quarter of \(2012 .\)

The level of air pollution in a certain city is proportional to the square of the population. Use calculus to estimate the percentage by which the air pollution level will increase if the population increases by \(5 \%\).

In its early phase, specifically the period 1984-1990, the AIDS epidemic could be modeled* by the cubic function $$ C(t)=-170.36 t^{3}+1,707.5 t^{2}+1,998.4 t+4,404.8 $$ for \(0 \leq t \leq 6\), where \(C\) is the number of reported cases \(t\) years after the base year \(1984 .\) a. Compute and interpret the derivative \(C^{\prime}(t)\). b. At what rate was the epidemic spreading in the year 1984 ? c. At what percentage rate was the epidemic spreading in 1984 ? In 1990 ?

The formula \(D=36 m^{-1.14}\) is sometimes used to determine the ideal population density \(D\) (individuals per square kilometer) for a large animal of mass \(m\) kilograms (kg). a. What is the ideal population density for humans, assuming that a typical human weighs about \(70 \mathrm{~kg}\) ? b. The area of the United States is about \(9.2\) million square kilometers. What would the population of the United States have to be for the population density to be ideal? c. Consider an island of area \(3,000 \mathrm{~km}^{2}\). Two hundred animals of mass \(m=30 \mathrm{~kg}\) are brought to the island, and \(t\) years later, the population is given by $$ P(t)=0.43 t^{2}+13.37 t+200 $$ How long does it take for the ideal population density to be reached? At what rate is the population changing when the ideal density is attained?

The population \(P\) of a bacterial colony \(t\) days after observation begins is modeled by the cubic function $$ P(t)=1.035 t^{3}+103.5 t^{2}+6,900 t+230,000 $$ a. Compute and interpret the derivative \(P^{\prime}(t)\). b. At what rate is the population changing after 1 day? After 10 days? c. What is the initial population of the colony? How long does it take for the population to double? At what rate is the population growing at the time it doubles?

The output at a certain factory is \(Q(L)=600 L^{2 / 3}\) units, where \(L\) is the size of the labor force. The manufacturer wishes to increase output by \(1 \%\). Use calculus to estimate the percentage increase in labor that will be required.

The output \(Q\) at a certain factory is related to inputs \(x\) and \(y\) by the equation $$ Q=x^{3}+2 x y^{2}+2 y^{3} $$ If the current levels of input are \(x=10\) and \(y=20\), use calculus to estimate the change in input \(y\) that should be made to offset an increase of \(0.5\) in input \(x\) so that output will be maintained at its current level.

You measure the radius \(r\) of a circle to be \(12 \mathrm{~cm}\) with an error no greater than \(3 \%\). Use calculus to estimate the error incurred by using this approximate value of \(r\) in the formula \(A=\pi r^{2}\) to compute the area of the circle.

Estimate what will happen to the volume of a cube if the length of each side is decreased by \(2 \%\). Express your answer as a percentage.

The output at a certain factory is \(Q=600 K^{1 / 2} L^{1 / 3}\) units, where \(K\) denotes the capital investment and \(L\) is the size of the labor force. Estimate the percentage increase in output that will result from a \(2 \%\) increase in the size of the labor force if capital investment is not changed.

The speed of blood flowing along the central axis of a certain artery is \(S(R)=1.8 \times 10^{5} R^{2}\) centimeters per second, where \(R\) is the radius of the artery. A medical researcher measures the radius of the artery to be \(1.2 \times 10^{-2}\) centimeter and makes an error of \(5 \times 10^{-4}\) centimeter. Estimate the amount by which the calculated value of the speed of the blood will differ from the true speed if the incorrect value of the radius is used in the formula.