Use the grid and a straight edge to make a rough estimate of the slope of the curve (in $y$ -units per $x$ -unit) at the points $P_{1}$ and $P_{2}$

(GRAPH CAN'T COPY)

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Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

$$y=4-x^{2}, \quad(-1,3)$$

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Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

$$y=(x-1)^{2}+1$$

Christopher S.

Numerade Educator

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

$$y=2 \sqrt{x}$$

Christopher S.

Numerade Educator

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

$$y=\frac{1}{x^{2}}, \quad(-1,1)$$

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Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

$$y=x^{3}, \quad(-2,-8)$$

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Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

$$y=\frac{1}{x^{3}}, \quad\left(-2,-\frac{1}{8}\right)$$

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Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.

$$f(x)=x^{2}+1, \quad(2,5)$$

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Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.

$$f(x)=x-2 x^{2}, \quad(1,-1)$$

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Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.

$$g(x)=\frac{x}{x-2}$$

Christopher S.

Numerade Educator

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.

$$g(x)=\frac{8}{x^{2}}$$

Christopher S.

Numerade Educator

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.

$$h(t)=t^{3}, \quad(2,8)$$

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Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.

$$h(t)=t^{3}+3 t$$

Christopher S.

Numerade Educator

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.

$$f(x)=\sqrt{x}, \quad(4,2)$$

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Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.

$$f(x)=\sqrt{x+1}$$

Christopher S.

Numerade Educator

Find the slope of the curve at the point indicated.

$$y=5 x-3 x^{2}, \quad x=1$$

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Find the slope of the curve at the point indicated.

$$y=x^{3}-2 x+7, x=-2$$

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Find the slope of the curve at the point indicated.

$$y=\frac{1}{x-1}, \quad x=3$$

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Find the slope of the curve at the point indicated.

$$y=\frac{x-1}{x+1}, x=0$$

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Growth of yeast cells In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number $P$ of cells present at hourly intervals. The number after $t$ hours is shown in the accompanying figure.

a. Explain what is meant by the derivative $P^{\prime}(5) .$ What are its units?

b. Which is larger, $P^{\prime}(2)$ or $P^{\prime}(3) ?$ Give a reason for your answer.

c. The quadratic curve capturing the trend of the data points (see Section 1.4 ) is given by $P(t)=6.10 t^{2}-9.28 t+16.43$ Find the instantaneous rate of growth when $t=5$ hours.

(FIGURE CAN'T COPY)

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Effectiveness of a drug On a scale from 0 to 1 , the effectiveness E of a pain-killing drug $t$ hours after entering the bloodstream is displayed in the accompanying figure.

a. At what times does the effectiveness appear to be increasing? What is true about the derivative at those times?

b. At what time would you estimate that the drug reaches its maximum effectiveness? What is true about the derivative at that time? What is true about the derivative as time increases

in the 1 hour before your estimated time?

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At what points do the graphs of the functions have a horizontal tangent lines?

$$f(x)=x^{2}+4 x-1$$

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At what points do the graphs of the functions have a horizontal tangent lines?

$$g(x)=x^{3}-3 x$$

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Find equations of all lines having slope -1 that are tangent to the curve $y=1 /(x-1)$

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Find an equation of the straight line having slope $1 / 4$ that is tangent to the curve $y=\sqrt{x}$

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Object dropped from a tower An object is dropped from the top of a $100-\mathrm{m}$ -high tower. Its height above ground after $t$ sec is $100-4.9 t^{2} \mathrm{m} .$ How fast is it falling 2 sec after it is dropped?

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Speed of a rocket At $t$ sec after liftoff, the height of a rocket is

$3 t^{2} \mathrm{ft} .$ How fast is the rocket climbing 10 sec after liftoff?

Christopher S.

Numerade Educator

Circle's changing area What is the rate of change of the area of a circle $\left(A=\pi r^{2}\right)$ with respect to the radius when the radius is $r=3 ?$

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Ball's changing volume What is the rate of change of the volume of a ball $\left(V=(4 / 3) \pi r^{3}\right)$ with respect to the radius when the radius is $r=2 ?$

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Show that the line $y=m x+b$ is its own tangent line at any point $\left(x_{1}, m x_{0}+b\right)$

Christopher S.

Numerade Educator

Find the slope of the tangent line to the curve $y=1 / \sqrt{x}$ at the point where $x=4$

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Does the graph of

$$

f(x)=\left\{\begin{array}{ll}

x^{2} \sin (1 / x), & x \neq 0 \\

0, & x=0

\end{array}\right.

$$

have a tangent line at the origin? Give reasons for your answer.

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Does the graph of

$$

g(x)=\left\{\begin{array}{ll}

x \sin (1 / x), & x \neq 0 \\

0, & x=0

\end{array}\right.

$$

have a tangent line at the origin? Give reasons for your answer.

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However, $y=x^{2 / 3}$ has no vertical tangent line at $x=0$ (see next figure):

$$

\begin{aligned}

\lim _{h \rightarrow 0} \frac{g(0+h)-g(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{2 / 3}-0}{h} \\

&=\lim _{h \rightarrow 0} \frac{1}{h^{1 / 3}}

\end{aligned}

$$

does not exist, because the limit is $\infty$ from the right and $-\infty$ from the Ieft.

Does the graph of

$$

f(x)=\left\{\begin{aligned}

-1, & x<0 \\

0, & x=0 \\

1, & x>0

\end{aligned}\right.

$$

have a vertical tangent line at the origin? Give reasons for your answer.

(GRAPH CAN'T COPY)

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We say that a continuous curve $y=f(x)$ has a vertical tangent line at the point where $x=x_{0}$ if the limit of the difference quotient is $\infty$ or - $\infty .$ For example, $y=x^{1 / 3}$ has a vertical tangent line at $x=0$ (see accompanying figure):

$$

\begin{aligned}

\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{1 / 3}-0}{h} \\

&=\lim _{h \rightarrow 0} \frac{1}{h^{2 / 3}}=\infty

\end{aligned}

$$

However, $y=x^{2 / 3}$ has no vertical tangent line at $x=0$ (see next figure):

$$

\begin{aligned}

\lim _{h \rightarrow 0} \frac{g(0+h)-g(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{2 / 3}-0}{h} \\

&=\lim _{h \rightarrow 0} \frac{1}{h^{1 / 3}}

\end{aligned}

$$

does not exist, because the limit is $\infty$ from the right and $-\infty$ from the Ieft.

Does the graph of

$$

U(x)=\left\{\begin{array}{ll}

0, & x<0 \\

1, & x \geq 0

\end{array}\right.

$$

have a vertical tangent line at the point (0,1)$?$ Give reasons for your answer

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=x^{2 / 3}$$

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=x^{4 / 5}$$

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=x^{1 / 3}$$

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=x^{3 / 5}$$

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=4 x^{2 / 5}-2 x$$

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=x^{5 / 3}-5 x^{2 / 3}$$

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=x^{2 / 3}-(x-1)^{1 / 3}$$

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=x^{1 / 3}+(x-1)^{1 / 3}$$

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=\left\{\begin{array}{ll}

-\sqrt{|x|}, & x \leq 0 \\

\sqrt{x}, & x>0

\end{array}\right.$$

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Graph the curves

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38

$$y=\sqrt{|4-x|}$$

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Use a CAS to perform the following steps for the functions.

a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$

b. Holding $x_{0}$ fixed, the difference quotient

$$

q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}

$$

at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.

c. Find the limit of $q$ as $h \rightarrow 0$

d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ and $1 .$ Graph them, together with $f$ and the tangent line, over the interval in part (a).

$$f(x)=x^{3}+2 x, \quad x_{0}=0$$

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Use a CAS to perform the following steps for the functions.

a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$

b. Holding $x_{0}$ fixed, the difference quotient

$$

q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}

$$

at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.

c. Find the limit of $q$ as $h \rightarrow 0$

d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ and $1 .$ Graph them, together with $f$ and the tangent line, over the interval in part (a).

$$f(x)=x+\frac{3}{x}, x_{0}=1$$

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Use a CAS to perform the following steps for the functions.

a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$

b. Holding $x_{0}$ fixed, the difference quotient

$$

q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}

$$

at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.

c. Find the limit of $q$ as $h \rightarrow 0$

d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ and $1 .$ Graph them, together with $f$ and the tangent line, over the interval in part (a).

$$f(x)=x+\sin (2 x), \quad x_{0}=\pi / 2$$

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Use a CAS to perform the following steps for the functions.

a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$

b. Holding $x_{0}$ fixed, the difference quotient

$$

q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}

$$

at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.

c. Find the limit of $q$ as $h \rightarrow 0$

d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ and $1 .$ Graph them, together with $f$ and the tangent line, over the interval in part (a).

$$f(x)=\cos x+4 \sin (2 x), \quad x_{0}=\pi$$

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