0
votes
0answers
28 views

True or false: differentiation. [on hold]

If the function $f(x,y): \mathbb{R}^2 \longrightarrow \mathbb{R}^3$ is differentiable at $(2,-1)$ with a tangent plane such as $z= 2x - 3y + 2$, then the function $g(x,y)= 3x - 2f(x,y) + 5$ is ...
0
votes
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 ...
1
vote
1answer
32 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?
0
votes
1answer
15 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) = ...
0
votes
2answers
84 views

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

Find $\dfrac{\mathrm d^2y}{\mathrm dx^2}$, as a function of $t$, for the given the parametric equations: $$\begin{align}x&=3-3\cos(t)\\y&=3+\cos^4(t)\end{align}$$ ...
1
vote
1answer
74 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 ...
0
votes
1answer
298 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 ...
1
vote
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 ...