Speed of a car The accompanying figure shows the time-to-distance graph for a sports car accelerating from a standstill.

a. Estimate the slopes of secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ , arranging them in order in a table like the one in Figure 2.6 . What are the appropriate units for these slopes?

b. Then estimate the car's speed at time $t=20 \mathrm{sec}$.

## Discussion

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## Recommended Questions

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$

$$

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

$$

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

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

\end{equation}

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

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

\end{equation}

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(b)$ an equation of the tangent line at $P$

$$y=x^{2}-5, \quad P(2,-1)$$

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

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

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

$y=x^{3}-3 x^{2}+4, \quad P(2,0)$

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

y=x^{3}-3 x^{2}+4, \quad P(2,0)

\end{equation}

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\begin{equation}

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

\end{equation}

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

y=\frac{x}{2-x}, \quad P(4,-2)

\end{equation}

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\begin{equation}

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

\end{equation}

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

y=\sqrt{x}, \quad P(4,2)

\end{equation}

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

y=x^{3}-12 x, \quad P(1,-11)

\end{equation}

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

y=7-x^{2}, \quad P(2,3)

\end{equation}

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

y=\sqrt{7-x}, \quad P(-2,3)

\end{equation}

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

y=\frac{1}{x}, \quad P(-2,-1 / 2)

\end{equation}

The slope of the tangent line to a curve is given by

$$f^{\prime}(x)=6 x^{2}-4 x+3$$

If the point $(0,1)$ is on the curve, find an equation of the curve.

In Exercises $7-18,$ use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ b) an equation of the tangent line at $P .$

\begin{equation}

y=x^{2}-5, \quad P(2,-1)

\end{equation}

Find an equation of the tangent line to the curve at the given point.

$ y = \sqrt {1 + x^3}, (2, 3) $

Find an equation of the tangent line to the curve at the given point.

$ y = x^3 - 3x + 1 $, $ (2, 3) $

Find the equation of the tangent line at the given point on each curve.

$x^{2}+y^{2}=25 ; \quad(-3,4)$

Find the equation of the tangent line at the given point on each curve.

$$

x^{2}+y^{2}=25 ; \quad(-3,4)

$$

Find an equation of the tangent to the curve at the point

corresponding to the given value of the parameter.

$$x=1+4 t-t^{2}, \quad y=2-t^{3} ; \quad t=1$$

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

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

Find an equation of the tangent line at the given point. If you have a CAS that will graph implicit curves, sketch the curve and the tangent line.

$$x^{3} y^{2}=-3 x y \text { at }(-1,-3)$$

Find an equation of the tangent to the curve at the point corresponding to the given values of the parameter

$ x = t^3 + 1 $, $ \quad y = t^4 + 1 $; $ \quad t = -1 $

Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent.

$ x = t^2 - t $, $ \quad y = t^2 + t + 1 $; $ \quad (0, 3) $

Use implicit differentiation to find an equation of the

tangent line to the curve at the given point.

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

Find an equation for (a) the tangent line to the curve at $P$ and (b) the horizontal tangent line to the curve at $Q$.(GRAPH CAN'T COPY)

Find an equation for (a) the tangent line to the curve at $P$ and (b) the horizontal tangent line to the curve at $Q$.(GRAPH CAN'T COPY)

Find an equation for the line tangent to the curve at the point defined by the given value of $t .$ Also, find the value of $d^{2} v / d x^{2}$ at this point.

$$x=\cos t, \quad y=\sqrt{3} \cos t, \quad t=2 \pi / 3$$

Find an equation for the line tangent to the curve at the point defined by the given value of $t .$ Also, find the value of $d^{2} v / d x^{2}$ at this point.

$$x=t, \quad y=\sqrt{t}, \quad t=1 / 4$$

Find an equation of the curve whose tangent line has a slope of

$f^{\prime}(x)=x^{2 / 3}$

given that the point $(1,3 / 5)$ is on the curve.

Find an equation of the curve whose tangent line has a slope of

$$f^{\prime}(x)=x^{2 / 3}$$

given that the point $(1,3 / 5)$ is on the curve.

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

$ 2(x^2 + y^2)^2 = 25(x^2 - y^2), (3, 1), $ (lemniscate)

Equation of a Tangent Line Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line.

$$

y=2 x-x^{3}, \quad \text { at }(1,1)

$$

Find an equation of the tangent line to the curve at the given point.

$ y = 4x - 3x^2 $, $ (2, -4) $

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

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

Compute the slope of the tangent line at the given point both explicitly (first solve for $y$ as a function of $x$ ) and implicitly.

$$x^{3} y-4 \sqrt{x}=x^{2} y \text { at }(2, \sqrt{2})$$

Find an equation of the tangent to the curve at the point

corresponding to the given value of the parameter.

$$x=t-t^{-1}, \quad y=1+t^{2} ; \quad t=1$$

Find an equation of the tangent to the curve at the point

corresponding to the given value of the parameter.

$$x=t-t^{-1}, \quad y=1+t^{2} ; \quad t=1$$

Let $f(x, y)=2 x+3 y-4 .$ Find the slope of the line tangent tothis surface at the point $(2,-1)$ and lying in the a. plane $x=2$ b. plane $y=-1 .$

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

Find an equation for the line tangent to the curve at the point defined by the given value of $t .$ Also, find the value of $d^{2} v / d x^{2}$ at this point.

$$x=-\sqrt{t+1}, \quad y=\sqrt{3 t}, \quad t=3$$