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Section 1

Derivatives and Rates of Change

A curve has equation $y=f(x)$ .(a) Write an expression for the slope of the secant line through the points $P(3, f(3))$ and $Q(x, f(x))$ (b) Write an expression for the slope of the tangent line at $P .$

Graph the curve $y=e^{x}$ in the viewing rectangles $[-1,1]$ by $[0,2],[-0.5,0.5]$ by $[0.5,1.5],$ and $[-0.1,0.1]$ by $[0.9,1.1] .$ What do you notice about the curve as you zoom in toward the point $(0,1) ?$

(a) Find the slope of the tangent line to the parabola $y=4 x-x^{2}$ at the point $(1,3)$(i) using Definition 2 (ii) using Equation 3(b) Find an equation of the tangent line in part (a).(c) Graph the parabola and the tangent line. As a check on your work, zoom in toward the point $(1,3)$ until the parabola and the tangent line are indistinguishable.

(a) Find the slope of the tangent line to the curve $y=x-x^{3}$ at the point $(1,0)$(i) using Definition 2 (ii) using Equation 3(b) Find an equation of the tangent line in part (a)(c) Graph the curve and the tangent line in successively smaller viewing rectangles centered at $(1,0)$ until the curve and the line appear to coincide.

5-8 Find an equation of the tangent line to the curve at the given point.$$y=4 x-3 x^{2}, \quad(2,-4)$$

5-8 Find an equation of the tangent line to the curve at the given point.$y=x^{3}-3 x+1,(2,3)$

5-8 Find an equation of the tangent line to the curve at the given point.$y=\sqrt{x}, \quad(1,1)$

5-8 Find an equation of the tangent line to the curve at the given point.$y=\frac{2 x+1}{x+2}, \quad(1,1)$

(a) Find the slope of the tangent to the curve $y=3+4 x^{2}-2 x^{3}$ at the point where $x=a$ .(b) Find equations of the tangent lines at the points $(1,5)$ and $(2,3) .$(c) Graph the curve and both tangents on a common screen.

(a) Find the slope of the tangent to the curve $y=1 / \sqrt{x}$ at the point where $x=a$ .(b) Find equations of the tangent lines at the points $(1,1)$ and $\left(4, \frac{1}{2}\right) .$(c) Graph the curve and both tangents on a common screen.

For the function $g$ whose graph is given, arrange the following numbers in increasing order and explain your reasoning:$$0 \quad g^{\prime}(-2) \quad g^{\prime}(0) \quad g^{\prime}(2) \quad g^{\prime}(4)$$

Find an equation of the tangent line to the graph of$y=g(x)$ at $x=5$ if $g(5)=-3$ and $g^{\prime}(5)=4$

If an equation of the tangent line to the curve $y=f(x)$ at the point where $a=2$ is $y=4 x-5,$ find $f(2)$ and $f^{\prime}(2)$

If the tangent line to $y=f(x)$ at $(4,3)$ passes through the point $(0,2),$ find $f(4)$ and $f^{\prime}(4)$

Sketch the graph of a function $f$ for which $f(0)=0$ $f^{\prime}(0)=3, f^{\prime}(1)=0,$ and $f^{\prime}(2)=-1$

Sketch the graph of a function $g$ for which $$\begin{array}{l}{g(0)=g(2)=g(4)=0, g^{\prime}(1)=g^{\prime}(3)=0} \\ {g^{\prime}(0)=g^{\prime}(4)=1, g^{\prime}(2)=-1, \lim _{x \rightarrow \infty} g(x)=\infty, \text { and }} \\ {\lim _{x \rightarrow-\infty} g(x)=-\infty}\end{array}$$

If $f(x)=3 x^{2}-x^{3},$ find $f^{\prime}(1)$ and use it to find an equation of the tangent line to the curve $y=3 x^{2}-x^{3}$ at the point $(1,2) .$

If $g(x)=x^{4}-2,$ find $g^{\prime}(1)$ and use it to find an equation of the tangent line to the curve $y=x^{4}-2$ at the point $(1,-1)$ .

(a) If $F(x)=5 x /\left(1+x^{2}\right),$ find $F^{\prime}(2)$ and use it to find an equation of the tangent line to the curve $y=5 x /\left(1+x^{2}\right)$ at the point $(2,2)$ .(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

(a) If $G(x)=4 x^{2}-x^{3},$ find $G^{\prime}(a)$ and use it to find equations of the tangent lines to the curve $y=4 x^{2}-x^{3}$ at the points $(2,8)$ and $(3,9)$ .(b) Illustrate part (a) by graphing the curve and the tangent lines on the same screen.

Find $$f^{\prime}(a)$$$$f(x)=3 x^{2}-4 x+1$$

Find $$f^{\prime}(a)$$$f(t)=2 t^{3}+t$

Find $$f^{\prime}(a)$$$f(t)=\frac{2 t+1}{t+3}$

Find $$f^{\prime}(a)$$$f(x)=x^{-2}$

Find $$f^{\prime}(a)$$$f(x)=\sqrt{1-2 x}$

Shown are graphs of the position functions of two runners, A and B, who run a 100 meter race and finish in a tie.(a) Describe and compare how the runners run the race.(b) At what time is the distance between the runners the greatest?(c) At what time do they have the same velocity?

If a ball is thrown into the air with a velocity of 40 $\mathrm{ft} / \mathrm{s}$ , its height (in feet) after $t$ seconds is given by $y=40 t-16 t^{2}$ . Find the velocity when $t=2$ .

If a rock is thrown upward on the planet Mars with a velocity of $10 \mathrm{m} / \mathrm{s},$ its height (in meters) after $t$ seconds is given by $H=10 t-1.86 t^{2}$(a) Find the velocity of the rock after one second.(b) Find the velocity of the rock when $t=a$ .(c) When will the rock hit the surface?(d) With what velocity will the rock hit the surface?

The displacement (in meters) of a particle moving in a straight line is given by the equation of motion $s=1 / t^{2}$ where $t$ is measured in seconds. Find the velocity of the particle at times $t=a, t=1, t=2,$ and $t=3$

Invasive species The Argentine ant is an invasive species in North America.The graph shows the cumulative number of counties in theUnited States that have been invaded by this species overtime.(a) Estimate the average rate of invasion between 1890 and $1920,$ between 1920 and $1960,$ and between 1960 and 2000 .(b) Estimate the instantaneous rate of invasion in $1940 .$

Population growth The table gives the US midyear population, in millions, from 1990 to $2010 .$$$\begin{array}{|c|c|c|c|c|c|}\hline t & {1990} & {1995} & {2000} & {2005} & {2010} \\ \hline P(t) & {249.6} & {266.3} & {282.2} & {295.8} & {308.3} \\ \hline\end{array}$$$$\begin{array}{c}{\text { (a) Find the average rate of population increase }} \\ {\text { (i) from } 1990 \text { to } 2000 \text { (ii) from } 1995 \text { to } 2000} \\ {\text { (iii) from } 2000 \text { to } 2005 \text { (iv) from } 2000 \text { to } 2010}\end{array}$$$$\begin{array}{l}{\text { (b) If } P(t) \text { is the population at time } t, \text { estimate and interpret }} \\ {\quad \text { the value of the derivative } P^{\prime}(2000) .}\end{array}$$

Viral load The table shows values of the viral load $V(t)$ in HIV patient 303 , measured in RNA copies/mL, $t$ days after ABT-5 38 treatment was begun.$$\begin{array}{|c|c|c|c|c|c|}\hline t & {4} & {8} & {11} & {15} & {22} \\ \hline V(t) & {53} & {18} & {9.4} & {5.2} & {3.6} \\ \hline\end{array}$$(a) Find the average rate of change of $V$ with respect to $t$over each time interval:$$\begin{array}{ll}{\text { (i) }[4,11]} & {\text { (ii) }[8,11]} \\ {\text { (iii) }[11,15]} & {\text { (iv) }[11,22]} \\ {\text { What are the units? }}\end{array}$$(b) Estimate and interpret the value of the derivative $V^{\prime}(11)$

Blood alcohol concentration Researchers measured the average blood alcohol concentration $C(t)$ of eight men starting one hour after consumption of 30 $\mathrm{mL}$ of ethanol (corresponding to two alcoholic drinks):$$\begin{array}{|c|c|c|c|c|c|}\hline t \text { hours) } & {1.0} & {1.5} & {2.0} & {2.5} & {3.0} \\ \hline C(t) & {(m g / m L)} & {0.33} & {0.24} & {0.18} & {0.12} & {0.07} \\ \hline\end{array}$$$$\begin{array}{ll}{\text { (i) }[1.0,2.0]} & {\text { (ii) }[1.5,2.0]} \\ {\text { (iii) }[2.0,2.5]} & {\text { (iv) }[2.0,3.0]} \\ {\text { What are the units? }}\end{array}$$(b) Estimate and interpret the value of the derivative $C^{\prime}(2)$

Let $D(t)$ be the US national debt at time $t .$ The table gives approximate values of the function by providing end of year estimates, in billions of dollars, from 1990 to $2010 .$ Interpret and estimate the value of $D^{\prime}(2000) .$$$\begin{array}{|c|c|c|c|c|c|}\hline t & {1990} & {1995} & {2000} & {2005} & {2010} \\ \hline D(t) & {3233} & {4974} & {5662} & {8170} & {14,025} \\ \hline\end{array}$$

Let $T(t)$ be the temperature (in ${ }^{\circ} \mathrm{F}$ ) in Seattle $t$ hours after midnight on May 7,2012 . The table shows values of this function recorded every two hours. What is the meaning of $T^{\prime}(12) ?$ Estimate its value.$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline t & 4 & 6 & 8 & 10 & 12 & 14 & 16 \\\hline T & 48 & 46 & 51 & 57 & 62 & 68 & 71 \\\hline\end{array}$$

A roast turkey is taken from an oven when its temperature has reached $185^{\circ} \mathrm{F}$ and is placed on a table in a room where the temperature is $75^{\circ} \mathrm{F}$ . The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.

A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater orless than the rate of change after an hour?

Bacteria population The number of bacteria after $t$ hours in a controlled laboratory experiment is $n=f(t)$ .(a) What is the meaning of the derivative $f^{\prime}(5) ?$ What are its units?(b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, $f^{\prime}(5)$ or $f^{\prime}(10) ?$ If the supply of nutrients is limited, would that affect your conclusion? Explain.

The cost of producing $x$ ounces of gold from a new gold mine is $C=f(x)$ dollars.(a) What is the meaning of the derivative $f^{\prime}(x) ?$ What are its units?(b) What does the statement $f^{\prime}(800)=17$ mean?(c) Do you think the values of $f^{\prime}(x)$ will increase or decrease in the short term? What about the long term? Explain.

The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of $p$ dollars per pound is $Q=f(p)$ .(a) What is the meaning of the derivative $f^{\prime}(8) ?$ What are its units?(b) Is $f^{\prime}(8)$ positive or negative? Explain.

Oxygen solubility The quantity of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences the oxygen content of water.)The graph shows how oxygen solubility $S$ varies as a function of the water temperature $T .$(a) What is the meaning of the derivative $S^{\prime}(T) ?$ What are its units?(b) Estimate the value of $S^{\prime}(16)$ and interpret it.

Swimming speed of salmon The graph at the right shows the influence of the temperature $T$ on the maximum sustainable swimming speed $S$ of Coho salmon.(a) What is the meaning of the derivative $S^{\prime}(T) ?$ What are its units?(b) Estimate the values of $S^{\prime}(15)$ and $S^{\prime}(25)$ and interpret them.