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

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

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

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$$$y=2-x^{3}, \quad P(1,1)$$

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

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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 .$$$y=5-x^{2}, \quad P(1,4)$$

a. $-2$b. $y=-x+6$

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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^{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=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=x^{3}-3 x^{2}+4, \quad P(2,0)\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=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=\frac{x}{2-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 $(\mathrm{b})$ 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 $($ 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 $P .$\begin{equation}y=\sqrt{7-x}, \quad P(-2,3)\end{equation}

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

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

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ 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=\frac{1}{x}, \quad P(-2,-1 / 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}

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 to the curve at the pointcorresponding 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 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)

Use implicit differentiation to find an equation of thetangent line to the curve at the given point.$x^{2 / 3}+y^{2 / 3}=4, \quad(-3 \sqrt{3}, 1)$ (astroid)

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.

Find the equation of the tangent line at the given point on each curve.$$e^{x^{2}+y^{2}}=x e^{5 y}-y^{2} e^{5 x / 2} ; \quad(2,1)$$

Find the equation of the tangent line at the given point on each curve. $$e^{x^{2}+y^{2}}=x e^{5 y}-y^{2} e^{5 x / 2} ; \quad(2,1)$$

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

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.$$y=2 \sqrt{x}$$

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)

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.$y=\frac{1}{x^{3}}, \quad\left(-2,-\frac{1}{8}\right)$

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 $P .$

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

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 $.

Find the slope of the curve at the point indicated.$y=5 x-3 x^{2}, \quad x=1$

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

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

\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=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=\frac{x}{2-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 $(\mathrm{b})$ 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 $($ 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 $P .$

\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 $(\mathrm{b})$ 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 $P$

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

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $($ 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=\frac{1}{x}, \quad P(-2,-1 / 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}

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 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 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)

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 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.

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

$$e^{x^{2}+y^{2}}=x e^{5 y}-y^{2} e^{5 x / 2} ; \quad(2,1)$$

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

$$

e^{x^{2}+y^{2}}=x e^{5 y}-y^{2} e^{5 x / 2} ; \quad(2,1)

$$

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

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

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

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

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)

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

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

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 $P .$

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.

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

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 $.

Find the slope of the curve at the point indicated.

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

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 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.

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