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CALCULUS AND ANALYSIS
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PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR-VALUED FUNCTIONS
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Differential equations
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Here's what students ask on Parametric equations
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-03-18
Write a short paragraph explaining this statement. Use the following example and your answers Does the particle travel clockwise or anticlockwise around the circle? Find parametric equations if the particles moves in the opposite direction around the circle. The position of a particle is given by the parametric equations
\(x = sin t, y = cos t\)
where 1 represents time. We know that the shape of the path of the particle is a circle.
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-03-09
Graph the plane curve defined by the parametric equations:
\(x = t^2 + 1, y = 3t, -2 \leq t \leq 2\)
.
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-03-05
1. A curve is given by the following parametric equations. x = 20 cost, y = 10 sint. The parametric equations are used to represent the location of a car going around the racetrack. a) What is the cartesian equation that represents the race track the car is traveling on? b) What parametric equations would we use to make the car go 3 times faster on the same track? c) What parametric equations would we use to make the car go half as fast on the same track? d) What parametric equations and restrictions on t would we use to make the car go clockwise (reverse direction) and only half-way around on an interval of [0, 2?]? e) Convert the cartesian equation you found in part “a” into a polar equation? Plug it into Desmos to check your work. You must solve for “r”, so “r = ?”
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-03-02
Given
\(w=2(\cos150∘+i\sin150∘)w=2(\cos150) and z=–\sqrt{2}+i\)
Find the polar form of z
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-03-01
Parametric equations and a value for the parameter t are given
\(x = (60 cos 30^{\circ})t, y = 5 + (60 sin 30^{\circ})t - 16t2, t = 2\)
. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t.
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-02-25
Solve.
\(\displaystyle\lim_{{{x}\rightarrow\infty}}{\left(\frac{{\sqrt[{{3}}]{{{x}^{{3}}-{2}{x}^{{2}}+{3}}}}}{{{2}{x}+{1}}}\right.}\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-02-24
Graph the plane curve defined by the parametric equations
\(x = 125 cos t and y = 125 sin t\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-02-21
Working with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation.
\(x=\ -7\ \cos\ 2t,\ y=\ -7\ \sin\ 2t,\ 0\ \leq\ t\ \leq\ \pi\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-02-12
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
\(x=3\ \ln(t),\ y=4t^{\frac{1}{2}},\ z=t^{3},\ (0,\ 4,\ 1)\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-02-12
Find parametric equations for the lines in The line through the origin parallel to the vector
\(2j + k\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-02-10
Find a set of parametric equations for the rectangular equation.
\(y = (x+3)^2 \backslash5\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-02-06
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
\(x=e^{-8t}\ \cos(8t),\ y=e^{-8t}\ \sin(8t),\ z=e^{-8t},\ (1,\ 0,\ 1)\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-02-02
Let
\(\displaystyle{f{{\left({x},{y}\right)}}}=-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\)
.
Find limit of
\(\displaystyle{f{{\left({x},{y}\right)}}}{a}{s}{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}{i}{)}{A}{l}{o}{n}{g}{y}{a}\xi{s}{\quad\text{and}\quad}{i}{i}{)}{a}{l}{o}{n}{g}{t}{h}{e}{l}\in{e}{y}={x}.{E}{v}{a}{l}{u}{a}{t}{e}\Lim{e}{s}\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{y}.{\log{{\left({x}^{{2}}+{y}^{{2}}\right)}}}\)
,by converting to polar coordinates.
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-02-02
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
\((-1,\ 0,\ 8)\)
Parallel to
\(v = 3i\ +\ 4j\ -\ 8k\)
The given point is
\((−1,\ 0,\ 8)\ \text{and the vector or line is}\ v = 3i\ +\ 4j\ −\ 8k.\)
(a) parametric equations (b) symmetric equations
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-01-19
Parametric Equations of a Hyperbola Eliminate the parameter 0 in the following parametric equations.
\(x = a tan 0 y = b sec\)
e
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-01-08
Investigation Consider the helix represented by the vector-valued function
\(r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >\)
(a) Write the length of the arc son the helix as a function of t by evaluating the integral
\(s=\ \int_{0}^{t}\ \sqrt{[x'(u)]^{2}\ +\ [y'(u)]^{2}\ +\ [z'(u)]^{2}\ du}\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2021-01-08
Parametric to polar equations Find an equation of the following curve in polar coordinates and describe the curve.
\(x = (1 + cos t) cos t, y = (1 + cos t) sin t, 0 \leq t \leq 2\pi\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2020-12-07
Equations of lines Find both the parametric and the vector equations of the following lines. The line through (0, 0, 1) in the direction of the vector
\(v = ⟨4, 7, 0⟩\)
Parametric equations, polar coordinates, and vector-valued functions
asked 2020-11-23
Find the coordinates of the point on the helix for arc lengths
\(s =\ \sqrt{5}\ and\ s = 4\)
. Consider the helix represented investigation by the vector-valued function
\(r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >\)
.
Parametric equations, polar coordinates, and vector-valued functions
asked 2020-11-17
Write a short paragraph explaining this statement. Use the following example and your answers How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. The position of a particle is given by the parametric equations
\(x = sin t, y = cos t\)
where 1 represents time. We know that the shape of the path of the particle is a circle.
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