Enroll in one of our FREE online STEM summer camps. Space is limited so join now!View Summer Courses

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

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

Need more help? Fill out this quick form to get professional live tutoring.

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

a. 2b. $y=2 x-7$

No transcript available

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}-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=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 $(\mathrm{b})$ 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=\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=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{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 $($ b) an equation of the tangent line at $P .$\begin{equation}y=x^{3}, \quad P(2,8)\end{equation}

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

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=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=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{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=\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}

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) $

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 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 for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.$y=4-x^{2}, \quad(-1,3)$

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.

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 of the tangent to the curve at the pointcorresponding to the given value of the parameter.$$x=1+4 t-t^{2}, \quad y=2-t^{3} ; \quad t=1$$

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 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) $

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

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

(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.

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

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

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 $P .$

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

Find the points on the curve $c(t)=\left(3 t^{2}-2 t, t^{3}-6 t\right)$ where the tangent line has slope $3 .$

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.$$x=\frac{1}{t+1}, \quad y=\frac{t}{t-1}, \quad t=2$$

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.$$x=t-\sin t, \quad y=1-\cos t, \quad t=\pi / 3$$

You must be logged in to like a video.

You must be logged in to bookmark a video.

Our educator team will work on creating an answer for you in the next 6 hours.

## Discussion

## Video Transcript

No transcript available

## 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}-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=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 $(\mathrm{b})$ 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=\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=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{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 $($ 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 $(\mathrm{b})$ 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=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=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{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=\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}

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) $

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

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

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.

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 of the tangent to the curve at the point

corresponding to the given value of the parameter.

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

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 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) $

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

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

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 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)

$$

(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.

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 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.

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

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

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

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 $P .$

If $ f(x) = 3x^2 - x^3 $, find $ f'(1) $ and use it to find an equation of the tangent line to the curve

$ y = 3x^2 - x^3 $ at the point $ (1, 2) $.

Find the points on the curve $c(t)=\left(3 t^{2}-2 t, t^{3}-6 t\right)$ where the tangent line has slope $3 .$

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.

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

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.

$$x=t-\sin t, \quad y=1-\cos t, \quad t=\pi / 3$$