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14 views

Which property can be used to derive a differential equation for a reparametrization

With $0\le t\le1$, two space curves given by: $$c_1(t)=(1,t,0)\quad\quad c_2(t)=(0,t,2t(1-t))$$ One of them, say $c_1$, must be reparametrized by $r(t)$ in order to minimize the area between the ...
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2answers
18 views

Parametric Eqn / Differentiation

Parametric eqns of a curve are $x = t + \frac{1}{t}$ , $y = t - \frac{1}{t}$, where $t$ cannot be $0$. At point $P$ on curve, $t = 3$ and the tangent to curve at $P$ meets the $x$-axis at $Q$. The ...
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0answers
17 views

Derivative of a parametrized vector on a nonfixed basis

Suppose a curve defined by a vector parametrized through the variable $u$, and expressed on a non-fixed base, like the polar coordinates base. You derive it with respect to that parameter. What ...
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1answer
36 views

can the derivative of a closed complex contour at any point be zero?

If C is a closed contour in the complex plane parametrized by z(t)=u(t)+i*v(t), can there be any point where z'(t)=0?
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1answer
17 views

Proving that two function coordinates of a parametric curve equals 1

I am having difficulty with this question, Note: This is not homework, It is from a practice test that I am using to study Consider the curve: $x(t) = \frac{1-t^2}{1+t^2}$ ; $y(t) = ...
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2answers
118 views

Parametric Equations: Find $\frac{\mathrm d^2y}{\mathrm dx^2}$.

Find $\dfrac{\mathrm d^2y}{\mathrm dx^2}$, as a function of $t$, for the given the parametric equations: $$\begin{align}x&=3-3\cos(t)\\y&=3+\cos^4(t)\end{align}$$ ...
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1answer
79 views

$\frac{dy/dt}{dx/dt} \text{ at } t = a \text{ or } \lim_{t \to a} \frac{dy/dt}{dx/dt} \text{?}$

Take an example of parametric equation: \begin{cases} x = t^3\\ y = t^6 \end{cases} Obviously the formula $\displaystyle \left. \frac{dy}{dx}=\frac{dy/dt}{dx/dt} \right.$ does not work at $t=0 ...
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1answer
306 views

Partial Derivatives and Chain Rule

I have a line $L\subset\mathbb{C}^n$ which is parametrized by $x_1=a_1t, x_2=a_2t,\dots, x_n=a_nt$, a function $f(x_1,\dots,x_n)$, and I want to look at the restriction of $f$ onto $L$. This is just ...
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2answers
80 views

For what values of $b$ does this function lack extrema?

I need to find the range of values of the parameter $b$ for which the function below has no extrema. $$ \frac{b}{3}\,8^x + (2) 4^x + (b+3) 2^x + b \ln(2)$$ In the beginning I thought it'd be ...