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Recent questions in Differential Calculus

MMDCCC50m 2022-11-19

Given a vector space $V$ , is it possible to endow it with two nonequivalent norms $‖ \cdot ‖_{1}$ and $‖ \cdot ‖_{2}$ such that any linear functional on $V$ is continuous in the sense of $‖ \cdot ‖_{1}$ if only if it is continuous in the sense of $‖ \cdot ‖_{2}$ ?
By nonquivalent norms I mean the induced topologies of the norms are different.

Emma Hobbs 2022-11-18

Find the integral:
$\int \tan^{2} x d x$

drogaid1d8 2022-11-17

What does continuity mean?

Frances Pham 2022-11-15

Optimization - What if solution of the lagrangian is the critical point of the constraint

Nicholas Hunter 2022-11-15

Find the limit:
$lim_{t \to 0} (\frac{e^{- 8 t} - 1}{t}, \frac{t^{7}}{t^{8} - t^{7}}, \frac{5}{7 + t})$

Cael Dickerson 2022-11-15

This is the given function
$f (x, y) = x^{4} - 2 (x^{2}) (y^{2}) + y^{4} + y^{3}$
how determine the nature of critical point (0,0) ?

ajumbaretu 2022-11-14

Сritical point $(0, 0)$ of the function $f (x, y) = x^{2} + 4 y^{3}$ . Since the Hessian is singular, we cannot conclude anything about the local extrema. However, a graph of this function clearly shows that (0,0) is neither a local max nor min. Show this without just referring to the graph. Further, is there a way to show that $(0, 0)$ is isolated or not isolated?

Sophie Marks 2022-11-14

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document $\begin{matrix} (\frac{\partial^{2} f (p)}{\partial x_{i} \partial x_{j}}) \end{matrix}$ $1 \leq i, j \leq n$
How can we deduce that f only has finitely many non-degenerate critical points by this function only has non-degenerate critical point? And how can we use the compact manifold’s properties to solve the question?

Alberto Calhoun 2022-11-13

If the first derivative is $0$ at a certain point, is that necessarily a critical point?

Karley Castillo 2022-11-12

How do stable, unstable, and centre subspaces correspond to stability of a critical point?

dannigurl21ck2 2022-11-12

Find the integral:
$\int \frac{\ln x}{x^{5}} d x$

Ricky Arias 2022-11-12

Let $f : M \to N$ be a smooth function between two smooth manifolds. Then $p \in M$ is a critical point if $d f_{p}$ is not surjective.

Simone Watts 2022-11-11

Why is $f^{'} (c) = does not exist$ does not exist a critical point?

Audrey Arnold 2022-11-11

Can we consolidate the definition of “critical point” in one-variable calculus?

Jenny Roberson 2022-11-11

Find the integral:
$\int \sqrt{7 x + 9} d x$

MMDCCC50m 2022-11-11

What is the 'stability' of a critical point of a ( $1$ dimensional) dynamical system?

mxty42ued 2022-11-10

Given a function, $f (x) = \sqrt{4 x - x^{2}}$ , what does it mean to find its critical points?
Assuming that the domain of the function in question is not explicitly stated, are we right to say that this function is well-defined only when $0 \leq x \leq 4$ ?

Rigoberto Drake 2022-11-08

Calculate:
$\frac{d}{d x} \sec x \ln x$

klesstilne1 2022-11-08

Let $f : U \to R$ , $U \subseteq R^{n}$ is open. if $\vec{a} \in U$ is local Maxima or local minima, then $\vec{a}$ is a critical point.
a proof assumes $(\nabla f) (\vec{a}) \neq {\vec{0}}^{t}$ , and therefore for small enough $h \in R$ , $h > 0$ we get:
$f (\vec{a} + h ((\nabla f) (\vec{a}))^{t}) > f (\vec{a})$
which is in contradiction to maxima.
why is have to be larger the $f (\vec{a})$

perlejatyh8 2022-11-08

let $f : R \to R$ be a continuous function such that $f (0) \leq 1$ and for all real $x$ , $f (x)^{2} - 3 f (x) + 2 \geq 0$ . Prove that $f (x) \leq 1$ for all real $x$ .
I think you would start by factorising to get $(f (x) - 2) (f (x) - 1) \geq 0$ and then $f (x) \geq 1$ but I'm not really sure where to go from there? I think maybe you can use the intermediate value theorem but I'm not exactly sure how. TIA

One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be solving equations with decimals, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.

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