Derivative Definition and 1000 Threads

1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...

The derivative of secx is $$\d{y}{x} secx =secx tanx $$ But if $$x = \frac{\pi}{3}$$, then $$secx = 2 $$ and the derivative of a constant is 0. And $$sec\frac{\pi}{3} tan\frac{\pi}{3}$$ is equal to $$\frac{3}{2}$$ So what is the derivative of $$secx$$ where $$x = \frac{\pi}{3}$$?

Suppose we have a general timelike congruence of curves with tangent vector field ##V##, then the standard decomposition of the covariant derivative in index form (see e.g. Hawking and Ellis' "Large scale structure of space and time" equation 4.17) is given by $$V_{a;b} = \omega_{ab} +...

Some context for my question: If you have a smooth manifold \mathcal{M} you can define tangent vectors to parametrized paths in the following way: If \mathcal{P}(s) is a parametrized path, then \frac{d}{ds} \mathcal{P}(s) = V where V is the differential operator that acts on scalar fields...

Homework Statement take the derivative of a(t) = b(t)c(t) Homework Equations chain rule The Attempt at a Solution Apply the chain rule: a'(t) = c(t)b'(t) + b(t)c'(t) Is this correct? Thank you.

Homework Statement y = x 2sinx Homework EquationsThe Attempt at a Solution Ok, so If I see an x in an exponent, I would want to use ln to 'bring it out', right? ln y = ln (x 2sinx) = ln x + ln 2sinx = ln x + sinx ln2 now I take the derivative : y'/y = 1/x + cosx ln2 multiply both sides...

Homework Statement Compute Derivative y = xx + sin(x) Homework EquationsThe Attempt at a Solution since I have x in the exponent (x^x), I multiply both sides by ln: ln y = ln xx + ln sin(x) the x in the exponent comes out into the front, right? y'/y = x ln x + ln sin (x) using product...

Hello, I am struggling with these two questions. I think here should be used a quotient rule, but I am not sure how to proceed. a) f(x)=sin$\frac{1}{x}$ b)g(x)=$\frac{1}{sinx}$ Can someone please help. Thanks

Hi. This is not a homework assignment. I am working to get an extrema on a graph that involves a bunch of functions and got stuck on one step: How to get the derivative of: \frac{dy}{dn} = \frac{nc(a+b)}{nc+a} I can't get "n" in a place where I recognize how to get the derivative of it. I...

Homework Statement I am given f(t) = e^-|t| and I found that F(w) = ##\sqrt{\frac{2}{\pi}}\frac{1}{w^2 + 1}## The question says to use the nth derivative property of the Fourier transform to find the Fourier transform of sgn(t)f(t), and gives a hint: "take the derivative of e^-|t|" I also...

[Note from mentor: This thread was originally posted in a non-homework forum, therefore it does not follow the standard homework template.] ------------------------------------------------ Hello. I have some homework to do. I need to make program that finds minimum/maximum of a function od 2...

Would it be a legitimate (valid) proof to use an \epsilon-\delta limit approach to prove the fundamental theorem of calculus? i.e. as the FTC states that if f is a continuous function on [a,b], then we can define a function F: [a,b]\rightarrow\mathbb{R} such that F(x)=\int_{a}^{x}f(t)dt Then F...

Homework Statement In what directions at the point (2, 0) does the function f(x, y) = xy have rate of change -1?D_{u}(f)(a,b) = \bigtriangledown f(a,b)\cdot (u_{1}, u_{2}) f(x,y) = xy (a,b) = (2,0). The Attempt at a Solution \frac{\partial f}{\partial x} = y \frac{\partial f}{\partial y} =...

Homework Statement Find the partial derivative of ## w = xe^\frac {y}{z} ##. Homework Equations N/A The Attempt at a Solution ## \frac{∂f}{∂x} = e^y/z ## ## \frac{∂f}{∂y} = \frac{xe^y/z}{z} ## ## \frac{∂f}{∂z} = (-yz^-2)(xe^yz^-1) ## Are theses correct? Thanks everyone.

In the book "Introduction to Mechanics" by K&K, an increment of a generic time-varying vector is split into two components, ##\Delta \vec{A} _{\perp}## and ##\Delta \vec{A}_{\parallel}##. Their magnitudes are approximated by: $$A \Delta \theta$$ and $$\Delta A$$ respectively. (Where ##\Delta...

Homework Statement D_{u}(f)(a,b) = \triangledown f(a,b)\cdot u D_{(\frac{1}{\sqrt2}, \frac{1}{\sqrt2})}(f)(a,b) = 3 \sqrt{2} where u = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2}) find \bigtriangledown f(a.b) Homework EquationsThe Attempt at a Solution first you change grad f into it's partial...

I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators \partial_{a} are dependent on the coordinate system one chooses and thus not naturally associated with the...

In chapter 1 of the book "Introduction to Mechanics" by Kleppner and Kolenkow, the derivative of a generic vector ##\vec{A}## is discussed in terms of decomposing an increment in ##\vec{A}##, ##Δ\vec{A}##, into two perpendicular vector vectors; one parallel to ##\vec{A}## and the other...

1. Given a function f(x,y) at (x0,y0). Find the two angles the directional derivative makes with the x-axis, where the directional derivative is 1. The angles lie in (-pi,pi]. 2. f(x,y) = sec(pi/14)*sqrt(x^2 + y^2) p0 = (6,6) 3. I use the relation D_u = grad(f) * u, where u is the...

The bulk modulus B = - V (∂P/∂V). At constant temperature the pressure is given by P= -∂U/∂V, where U is the total energy. We can write B in terms of the energy per particle u = U/N and volume per particle v = V/N : B = v...

Homework Statement Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) \tag 1$$ Homework Equations Below The Attempt at a Solution**MY PROOF:** If $(1)$ is...

Hello, I have this exercise that I can't get the right answer. I have to find derivative of g(x)= (4${x}^{2}$-2x+1)${e}^{x}$ So, what is did is g$^{\prime}$=(8x-2)${e}^{x}$+(4${x}^{2}$-2x+1)${e}^{x}$ My Prof said it is wrong... I am not sure if I have to multiply the brackets or what I did...

Homework Statement I recently searched around SE, and found: http://math.stackexchange.com/questions/1142546/how-to-solve-this-derivative-of-f-proofHomework Equations Below The Attempt at a Solution The answer is interesting. "A function given that $$f(x)=f''(x)+f'(x)g(x)$$ could be an...

Homework Statement Find (∂z/∂x) of 6xyz Homework Equations N/a The Attempt at a Solution The correct answer is 6xy(∂z/∂x) but I would like proof of it. I got something different when I tried taking the partial derivative. 6xyz = 6x(yz) = Multiplication rule for derivatives 6(∂x/∂x) +...

Whenever the second order derivative of any physical quantity is related to its second order space derivative a wave of some sort must travel in a medium, why this is so?

i just started with calculus. There was this question my teacher asked us d/dx (x²) = 2x ... eq 1 now we can write 2² = (2+2) 3² = (3+3+3) 4²=(4+4+4+4) . . . n² = (n+n+n+n+...)n times so here d/dx (x²) = d/dx (x+x+x+...)x times so ⇒ d/dx (x) +d/dx(x) +...(x times) = 1+1+1+...(x times) = x ⇒d/dx...

Hi, I'm using partial derivatives to calculate propagation of error. However, a bit rusty on my calculus. I'm trying to figure out the partial derivative with respect to L of the equation: 2pi*sqrt(L/g) (Yep, period of a pendulum). "g" is assumed to have no error. I know I can use the...

Homework Statement Dear Mentors PF Helpers, Here's my question: I see it from my textbook with it solutions copied down below. Wonder is there another way to do it. Thank you for your time.Homework Equations [/B]The Attempt at a Solution

I've attached an image to this post. It essentially shows the equation for the first partial derivative using chain rule, which makes sense. What I'm confused with is how the second partial derivative was formulated. It seems they've simply squared the first partial derivative to find the second...

How do I prove that the parity operator Af(x) = f(-x) commutes with the second derivative operator. I am tempted to write: A∂^2f(x)/∂x^2 = ∂^2f(-x)/∂(-x)^2 = ∂^2f(-x)/∂x^2 = ∂^2Af(x)/∂x^2 But that looks to be abuse of notation..

We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be defined as differentiable. However in the case of 1 independent variable, is it possible for a...

Homework Statement if $$y = \frac{2x^5-3x^3+x^2}{x^3}$$ then $$\frac{dy}{dx} =$$ Homework Equations if $$f(x) = x^n$$ then $$f'(x) = nx^{n-1}$$ The Attempt at a Solution $$\frac{2x^5-3x^3+x^2}{x^3} = \frac{2x^5}{x^3} - \frac{3x^3}{x^3} + \frac{x^2}{x^3}$$ $$ f'(\frac{2x^5-3x^3+x^2}{x^3}) =...

Hello, I try to apprehend the notion of covariant derivative. In order to undertsand better, here is a figure on which we are searching for express the difference \vec{V} = \vec{V}(M') - \vec{V}(M) : In order to evaluate this difference, we do a parallel transport of \vec{V}(M') at point...

Homework Statement What is the MRS of the quasilinear utility function U(q1, q2) = u(q1) + q2 ? Homework Equations MRS = - dU1/dU2 The Attempt at a Solution [/B]dU2 is 1 but I am unsure how to approach taking the derivative of u(q1). I have tried the answer as -dU and -dU * dq1...

Homework Statement Determine the value(s) of k such that y=-5k is the equation of the tangent line on the graph of F(x) = -x^2 + 4kx + 1 Homework Equations n/a The Attempt at a Solution not sure where to start this problem; but i understand some fundamentals here. I believe that the tangent...

f(x) = 3x^3 + 3x^2+ 2x + 1 ,a = 3 formal is Homework is due tonight and this is the only problem i can't solve Your suppose to 3= 3x^3 + 3x^2 + 2x + 1 , solve for xThe find the derivative of y= 3x^3 + 3x^2 + 2x + 1 , then plug x into that and put it under 1.

As we can not meaningfully compare a vector at 2 points acted upon by this operator , because it does not take into account the change due to the coordinate system constantly changing, I conclude that the elementary differential operator must describe a change with respect to space-time, How do...

Homework Statement FIGURE 5 shows a proportional plus derivative controller that has aproportional band of 20% and a derivative action time of 0.1 minutes. Construct the shape of the output waveform for the triangular input waveform shown, if the input rises and falls at the rate of 4 units...

I would like to know how to correctly define and classify turning points using elementary calculus. The points I wish to clarify are maxima, minima, inflection points and saddle points. So I am aware of the basic info available everywhere, such as that a point is a maximum if and only if the...

Homework Statement For f(x) = abs(x^3 - 9x), does f'(0) exist? The Attempt at a Solution [/B] The way I tried to solve this question was to find the right hand and left hand derivative at x = 0. Right hand derivative = (lim h--> 0+) f(h) - f(0) / h = (lim h--> 0+) abs(h^3 - 9h) / h...

I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function? If an integral is the area under a curve, why do we even have to find the anti derivative...

Homework Statement Given position function r(t) and r'(t) = c X r(t), where c is some constant vector, describe the path of the particle. In other words, describe r(t). Homework Equations // The Attempt at a Solutiona) r'(t) points in the direction of motion. If we can understand how r'(t)...

Homework Statement Finding the derivative of an inverse trigonometric functionHomework Equations [/B] *This is the problem*The Attempt at a Solution [/B] In my textbook, Single Variable Essential Calculus, Second Edition, by James Stewart, the derivative rules for the inverse trigonometric...

Homework Statement I would appreciate feedback on the following two problems: (1) For a given operator A with no explicit time dependence I am asked to show that d/dt(eAt)=A(eAt) (2) A free wave packet of width Δx is traveling at a constant velocity v0=p0/m. I am asked to estimate the...

Homework Statement I want to find the partial derivatives in the point (0,0) of the function f:\mathbb R^2\rightarrow\mathbb R f(x,y):= \begin{cases} 0 & \text{if } (x,y) = (0,0) \\ \frac{y^5}{2x^4+y^4} & otherwise \end{cases} Homework Equations Our definition of the partial derivatives in...

d(x^^n)/dx = ?

I'm trying to take the derivative of [x/x^2+1]^3. Where do I start?

Homework Statement Suppose that an amount function ## a(t) ## is differentiable and satisfies the property ## a(s + t) = a(s) + a(t) − a(0) ## for all non-negative real numbers ## s ## and ## t ##. (a) Using the definition of derivative as a limit of a difference quotient, show that ## a'(t) =...

Homework Statement Find the derivative of f(X). f(X) = transpose(a) * X * b where: X is nxn a and b are n x 1 ai is the i'th element of a Xnm is the element in row n and column m let transpose(a) = aT let transpose(b) = bT Homework Equations I tried using the product rule...

Homework Statement [/B] I'm supposed to find the derivative of 2^x using the definition of a derivative. I am really confused as to how I can factor out the h. Homework Equations y=2^x The Attempt at a Solution limit as h->0 in all of these, I don't want to write it out because it's going to...