Tagged Questions

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Wrong answer within 'Calculus Solution Manual, Michael Spivak, 3rd ed'

I have a problem with the answer provided in the solution manual of Calculus, Michael Spivak, 3rd ed, The Problem: Consider a hyperbola, where the difference of the distance between the two foci is ...
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How to calculate the polar arc length of the entire cardioid $r=a(1-\cos\theta)$

I'm having a bit of an issue calculating the arc length of $r = a(1-\cos\theta)$. I'll begin by listing the steps I made in my attempt to solve this exercise. We know that the arc length formula ...
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Polar to phasor

Let's say that there is a polar equation: -2400 + 8320j To convert this polar equation to phasor form, should the negative be considered when trying to find the angle? Would the angle be +73.91 or ...
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How to solve a complex equation $w^4 = \sqrt{3} -i$

$z = \sqrt{3} -i$ How do I solve a complex equation $w^4 = \sqrt{3} -i$ I know that I first have to rewrite z to polar format which I have done as $z = 2(cos (-π/6) + sin (-π/6))$ but I do not know ...
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Path of an ellipse

A path is described by the position vector $\mathbf{r}$: $$\mathbf{r}=a\cos(\omega t)\mathbf{\hat{i}}+b\sin{\omega t}\mathbf{\hat{j}}$$ I am asked to show that the path is the ellipse in the form of: ...
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Parametrization of a curve in polar coordinates

I'm trying to change this parametrics equations to polar coordinates $$X(t) = 2\cos(t) - \sin(2t) \\ Y(t) = 2\sin(t) - \cos(2t)$$ What i tryed to do was raise the two equations squared, sum ...
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Im trying to figure out but for some reason I dont know how to...could someone please tell me how did they get this answer from the solution manual....they skipped steps so I have no idea
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Find the area of the Rose's petal.

If a Rose leaf is described by the equation $r = \sin 3\theta$, find the area of one petal.
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Cartesian and Polar Coordinate

I should give the Cartesian Coordinates $(x,y)\in \mathbb{R\times R}$ and Polar Coordinates $(r,\varphi)\in R^+\times [0,2\pi)$ of the following Complex Numbers: a) $z_{1}=-i$ b) $z_{2}=\sqrt{3}+i$ ...
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Polar coordinates - issue with direction denoted by angle

Convert $1-\sqrt{3}i$ to polar coordinates $(r,\varphi)$. I started by computing $r=|1-\sqrt{3}i|=\sqrt{1^2+\sqrt{3}^2}=\sqrt{4}=2$. When I tried to compute the angle I did something like ...
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Cardioid given by the polar equation $r = 1 − \cos(\theta)$

Let $C$ be the cardioid given by the polar equation $r = 1 − \cos(\theta)$ , $−\pi \le \theta \le \pi$. (a) Find the intersection points of the curve with the line $\theta = \pi/4$. (b) Find the ...
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(Calculus 3) Having trouble finding the polar equation of a hyperbola.

Eccentricity e=sqrt(2), and one vertex is located at (2,0). I do know that if the vertex is located at (2,0), then the directrix is 2 units from the vertex. I am not sure how to find the location of ...
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Magnitude of average velocity..

A particle is at the position with Cartesian components $(5.0 m, 4.0 m)$ and $5.0 s$ later it is at the position with components $(8.0 m, 0.0 m)$. What is the magnitude of its average velocity? I ...
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The point C has coordinates (2,k) and the distance from A to C is 5. Find the 2 possible values of the constant k

The coordinates of A are (6, -2) I have managed to do every other question in this section but this one, my brother tried to help but just drew a graph and the answers without explaining it, i dont ...
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Find the length of a curve

Find the length of the following curve: $r(t) = e^{-t} \sin(t)+e^{-t}\sin(t) i$ for $0 \leq t\leq 1$. Any ideas?
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Find the equation in polar coordinate form for a straight line through the points with polar coordinates (4,0) and (4,π/3).

Find the equation in polar coordinate form for a straight line through the points with polar coordinates $(4,0)$ and $(4,π/3)$. Here's my steps: 1.Write the two points in cartesian coordinates: the ...
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Maximum point of a polar function

I have a curve C with polar equation $$r^2 = a^2\cos{2\theta}$$ And I am looking to find the length $x$ when $r=max$ Judging from the equation: $$r = \sqrt{a^2\cos{2\theta}}$$ R will be maximum ...
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Find the area of the region determined by two curves

Find the area of the region $R$ given by two curves. So the region $R$ describes the area that is common between the two curves: \begin{align*} \text{Function 1: } r&= 2\sin(\theta)\\ ...
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Is $r=2\cos(\theta)$ a one-petal polar function?

I'm currently learning about polar functions and their graphs in precalculus, and one of the questions on my homework is to identify the shape of the function $r=2\cos(\theta)$. We were taught that ...
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Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation.

Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation. Let $t= \theta$, $r=5$, $x=r\cos t$, $y=r\sin t$. I began with $x=5\cos t-5=5(\cos t-1)$ and $y=5\sin t$. Is that ...
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Laplacian in polar coordinates

I am stuck with an exercise that requires me to find the Laplacian $\Delta u=(D_x^2u+D_y^2u)$ of a 2d-function $u$ in polar coordinates (in the standard Euclidean plane). I found the following ...
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Transform integral into polar coordinates

At university we are given a voluntary hand in in the use of maple/matlab, in that regard I have a double integral I am in dire need to compute, using first cartesian then polarcoordinates. ...
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Polar Coordinates

It's been ages since i did any coordinate conversions, and typically i have these two which i just can't manage to solve by myself. I want to express the circle $x^{2}+y^{2}<4, x<0$ The Area: ...
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Help needed with partial derivatives and polar coordinates, missing term.

I have a missing $\frac{1}{r}\partial_r$ -term (notice the question mark) but cannot see why, could someone hint where I am doing mistake.
The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...