Find parametric equations for the tangent line to the curve

podnescijy 2021-11-20 Answered
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
\(\displaystyle{x}={1}+{2}\sqrt{{{t}}},\ {y}={t}^{{3}}-{t},\ {z}={t}^{{3}}+{t};\ {\left({3},{0},{2}\right)}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Heack1991
Answered 2021-11-21 Author has 1095 answers

We're given parametric equation \(\displaystyle{x}={1}+{2}\sqrt{{{t}}},\ {y}={t}^{{3}}-{t}\) and \(\displaystyle{z}={t}^{{3}}+{t}\) and we're asked to solve for the parametric equations of the tangent line to the curve at the point (3,0,2)
Given parametric equations, we know that the vector equatiom tr(t) is equal to
\(\displaystyle{r}{\left({t}\right)}={<}{1}+{2}\sqrt{{{t}}},{t}^{{3}}-{t},{t}^{{3}}+{t}{>}\)
Solve for \(\displaystyle{r}'{\left({t}\right)}\) by differentiating each of the components of r(t) with respect to t
\(\displaystyle{r}'{\left({t}\right)}={<}{\frac{{{1}}}{{\sqrt{{{t}}}}}},{3}{t}^{{2}}-{1},{3}{t}^{{2}}+{1}{>}\)
The parameter value corresponding to \(\displaystyle{\left({3},{0},{2}\right)}\) is t=1. Plug in t=1 into \(\displaystyle{r}'{\left({t}\right)}\) to solve for \(\displaystyle{r}'{\left({1}\right)}\)
\(\displaystyle{r}'{\left({1}\right)}={<}{\frac{{{1}}}{{\sqrt{{{1}}}}}},{3}{\left({1}\right)}^{{2}}-{1},{3}{\left({1}\right)}^{{2}}+{1}{>}\)
\(\displaystyle={<}{1},{2},{4}{>}\)
Recall from the textbook that parametric equations for a line through the point \(\displaystyle{\left({x}_{{0}},{y}_{{0}},{z}_{{0}}\right)}\) and parallel to the direction vector \(\displaystyle{<}{a},{b},{c}{>}\) are
\(\displaystyle{x}={x}_{{0}}+{a}{t}\ {y}={y}_{{0}}+{b}{t}\ {z}={z}_{{0}}+{c}{t}\)
Substitute \(\displaystyle{\left({x}_{{0}},{y}_{{0}},{z}_{{0}}\right)}={\left({3},{0},{2}\right)}\) and \(\displaystyle{<}{a},{b},{c}\ge{<}{1},{2},{4}{>}\) into x, y and z, respectively to solve for the parametric equations of the tangent line to the curve
\(\displaystyle{x}={3}+{\left({1}\right)}{t}={3}+{t}\ {y}={\left({0}\right)}+{\left({2}\right)}{t}={2}{t}\ {z}={\left({2}\right)}+{\left({4}\right)}{t}={2}+{4}{t}\)

Have a similar question?
Ask An Expert
0
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-09-03
Find the tangent line(s) to the parametric curve given by
\(\displaystyle{x}={t}^{{5}}-{4}{t}^{{3}}\)
\(\displaystyle{y}={t}^{{2}}\)
asked 2021-09-10

Determine the line integral along the curve C from A to B. Find the parametric form of the curve C
Use the vector field:
\(\overrightarrow{F}=[-bx\ cy]\)
Use the following values: \(a=5,\ b=2,\ c=3\)

asked 2021-09-04
Sketch the parametric curve for the following set of parametric equations. Clearly indicate direction of motion
1)\(\displaystyle{x}={5}{\cos{{t}}}\)
2)\(\displaystyle{y}={2}{\sin{{t}}}\)
3)\(\displaystyle{0}\ge{t}\ge{2}\pi\)
asked 2021-11-21
Find parametric equations and symmetric equations for the line.
The line through (-6, 2, 3) and parallel to the line \(\displaystyle{\frac{{{1}}}{{{2}{x}}}}={\frac{{{1}}}{{{3}{y}}}}={z}+{1}\)
asked 2021-11-12
Find a vector equation and parametric equations for the line segment that joins P to Q.
\(\displaystyle{P}{\left({1},-{1},{8}\right)},\ {Q}{\left({8},{2},{1}\right)}\)
asked 2021-11-14
Find parametric equations for the line through the point (0,1,2) that is perpendicular to the line x=1+t, y=1-t, z=2t and intersects this line.
asked 2021-09-15

Find sets of parametric equations and symmetric equations of the line that passes through the given point and is parallel to the given vector or line. (For each line, write the direction numbers as integers.)
Point: (-3, 4, 5)
Parallel to: \(\displaystyle\frac{{{x}-{1}}}{{2}}=\frac{{{y}+{1}}}{ -{{3}}}={z}-{5}\)
(a) Parametric equations
(b) Symmetric equations

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...