## 17Calculus Parametric Equations - Slope & Tangent Lines

##### 17Calculus

On this page we discuss calculating slope and tangent lines of curves that are described by parametric equations.

### Resources

Topics You Need To Understand For This Page

basics of parametric equations

derivatives of parametric equations

Related Topics and Links

Wikipedia - Parametric Derivative

To find the equation of a tangent line to a graph given by a set of parametric equations, we need to be able to find the slope by calculating the derivative $$dy/dx$$ using the parametric derivative $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \text{ where } dx/dt \neq 0$ For horizontal tangent lines, the slope $$dy/dx$$ is zero, so we need $$dy/dt = 0$$ and $$dx/dt \neq 0$$. For vertical tangent lines, the slope is undefined, which means that $$dx/dt = 0$$ when $$dy/dt \neq 0$$. In the case where both $$dx/dt = 0$$ and $$dy/dt = 0$$ at the same point, we need to handle that case separately, since nothing can be concluded from $$dy/dx = 0/0$$, which is indeterminate.

Once you have found the slope, you can easily find the equation of a tangent line. Go to the Tangent Lines page for more information.

Practice

Unless otherwise instructed, find the equation of the tangent line to these parametric curves at the given points.

$$x=4t$$, $$y=3t^2+2$$; $$t=2$$.

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=4t$$, $$y=3t^2+2$$ at $$t=2$$.

Solution

### The Organic Chemistry Tutor - 3501 video solution

Log in to rate this practice problem and to see it's current rating.

$$x=t^2-5$$, $$y=t^2-4t$$; $$(-4,-3)$$

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=t^2-5$$, $$y=t^2-4t$$ at $$(-4,-3)$$.

Solution

### The Organic Chemistry Tutor - 3502 video solution

Log in to rate this practice problem and to see it's current rating.

$$x=8\cos\theta$$, $$y=6\sin\theta$$; $$\theta = \pi/3$$

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=8\cos\theta$$, $$y=6\sin\theta$$ at $$\theta = \pi/3$$.

Solution

### The Organic Chemistry Tutor - 3503 video solution

Log in to rate this practice problem and to see it's current rating.

$$x=2t^2+1$$, $$y=3t^3+2$$; $$t=1$$.

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=2t^2+1$$, $$y=3t^3+2$$ at $$t=1$$.

Solution

### Krista King Math - 54 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

$$x=3(t-\sin(t))$$, $$y=3(1-\cos(t))$$; $$t=\pi/2$$.

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=3(t-\sin(t))$$, $$y=3(1-\cos(t))$$ at $$t=\pi/2$$.

Solution

### Krista King Math - 55 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

$$x=t\cos(t)$$, $$y=t\sin(t)$$; $$t=\pi$$

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=t\cos(t)$$, $$y=t\sin(t)$$ at $$t=\pi$$.

Solution

### Krista King Math - 467 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

$$x=3t^2-t$$, $$y=\sqrt{t}$$; $$t=4$$.

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=3t^2-t$$, $$y=\sqrt{t}$$ at $$t=4$$.

Solution

### PatrickJMT - 1371 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

$$x = 2\cos(3t) - 4\sin(3t)$$, $$y = 3\tan(6t)$$ at $$t = \pi/2$$

Problem Statement

Find the equation of the tangent line to the parametric curve $$x = 2\cos(3t) - 4\sin(3t)$$, $$y = 3\tan(6t)$$ at $$t = \pi/2$$ .

Solution

The solution can be found on this page.

Log in to rate this practice problem and to see it's current rating.

$$x = t^2 - 2t - 11$$, $$y = t(t-4)^3 - 3t^2(t-4)^2 + 7$$ at $$(-3, 7)$$

Problem Statement

Find the equation of the tangent line to the parametric curve $$x = t^2 - 2t - 11$$, $$y = t(t-4)^3 - 3t^2(t-4)^2 + 7$$ at $$(-3, 7)$$ .

Solution

The solution can be found on this page.

Log in to rate this practice problem and to see it's current rating.

Find the values of $$t$$ that will have horizontal or vertical tangent lines for the parametric equations $$x = t^5 - 7t^4 - 3t^3$$, $$y = 2\cos(3t) + 4t$$

Problem Statement

Find the values of $$t$$ that will have horizontal or vertical tangent lines for the parametric equations $$x = t^5 - 7t^4 - 3t^3$$, $$y = 2\cos(3t) + 4t$$

Solution

The solution can be found on this page.

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.