Second derivative Definition and 178 Threads

Hello, I have two problems. I'm going through the Classical Theory of Fields by Landau/Lifshitz and in Section 32 they're deriving the energy-momentum tensor for a general field. We started with a generalized action (in 4 dimensions) and ended up with the definition of a tensor...

This is a problem from My book which I have hard understanding what they are asking for, I am pretty confused on the question would like to have help! Second derivate test works as follows: If f (c) = 0 and f'' (c)> 0 Then c is a local min point for function f (a) Show that c need not be a...

So this is a theoretical question I am not sure about with partial derivatives. Say we have function x(i,j) with a=dx/di and b=dx/dj Now is this logical when the cross derivatives are required to be equal? db/da --> substitute b = d^2x/(dadj) substitute da = d^2x/((d^2x/di)dj)...

Suppose ##f^{\prime\prime}## is continuous on an open interval that contains x = c 1. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)<0##, then ##f## has local maximum at x = c. 2. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)>0##, then ##f## has local minimum at x = c. 3. If...

The graph of a differentiable function y=f(x) is 1. concave up on an interval I if f' is increasing on I. 2. concave down on an interval I if f' is decreasing on I. Let y=f(x) is twice differentiable on an interval I 1. If f'' > 0 on I, the graph of f over I is concave up. 2. If f'' <...

Homework Statement Just started getting introduced to calculus and a couple applications. After I've found the stationary point i understand that i can put the x value into the second derivative to find if its a maximum or minimum point. i.e 12x-2x2 ƒ'(x)= 12-4x 12-4x=0 x=3 so...

Homework Statement Find the second derivative of sin y+cos y=x, giving your answer in terms of x. Homework Equations Implicit derivatives The Attempt at a Solution ##\sin y+\cos y=x## ##\cos y \frac{dy}{dx} -\sin y \frac{dy}{dx}=1## ##\frac{dy}{dx}=\frac{1}{\cos y-\sin y}##...

Hello all, I have a problem with second derivatives and chain rule. I am working on the question attached (sorry, my Latex editor wasn't working...) I need to find F'(1) and F''(1). I managed to solve F'(1), but I can't figure out F''(1). In the second image attached, you can see the solution...

How does one derive the second derivative test for three variables? It's clear that D(a,b) = fxx * fyy - (fxy)^2 AND fxx(a,b) Tells us almost all we need to know about local maxima and local minima for a function of 2 variables x and y, but how do I make sense of the second directional...

Hi, if oyu look at question 16b in the link below in order to get the second derivative wrt to t they take the square of the first derivative. I don't get it, how does multiplying the first derivative by itself get you the second derivative...

Homework Statement Find d2/dx2 0∫x (1∫sint√(1+u^4)du)dtHomework Equations The Attempt at a Solution Initially I treated this problem as the second derivative of a double integral and thus quickly found myself at the result cosx√1+sin4x, by the fundamental theorem of calculus. However I...

Homework Statement . Let ##f:[a,b] \to [\alpha,\beta]## be a bijective function of class ##C^2## with an inverse function also of class ##C^2## ##g:[\alpha,\beta] \to [a,b]##. a)Calculate ##g''(x)## for every ##x \in (\alpha,\beta)## in terms of ##f## and its derivatives. b)If ##f'(x)>0##...

Hey guys. I am having some trouble visualizing one aspect of the Second derivative test in the 2 variable case (related to #3 below). Essentially, what does the curve look like when f_{xx}f_{yy} > 0, BUT f_{xx}f_{yy} < [f_{xy}]^{2}? To be more detailed, if the function is f(x,y), H(x,y) is the...

Given a function f(x(t, s) y(t, s)), if is possible to compact \frac{∂f}{∂t}=\frac{∂f}{∂x} \frac{∂x}{∂t}+\frac{∂f}{∂y} \frac{∂y}{∂t} by \frac{df}{dt}=\bigtriangledown f\cdot \frac{d\vec{r}}{dt} So, analogously, isn't possible to compact the sencond derivate \frac{\partial^2 f}{\partial s...

Is it possible to formulate the second derivate trough of a tree diagram, as we do with a first derivative? If yes, how do it? \frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}\frac{\partial v}{\partial t}+\frac{\partial f}{\partial...

Homework Statement Find the second derivative of the function: f(x)= x^(2/3) (6-x)^(1/3) Homework Equations The chain, product and quotient rules The Attempt at a Solution I have found the first derivative and checked my solution: ′()= 4− / ^(1/3) (6−)^(2/3) The final...

How do I find the second derivative of the function: f(x)= x^(2/3) (6-x)^(1/3) I have found the first derivative and checked my solution: ′()= 4− / ^(1/3) (6−)^(2/3) The final solution is supposed to be: ''()= -8 / ^(4/3) (6−)^(5/3) I know almost all the steps but I couldn't...

Homework Statement Let f(x,y) be a differentiable function with x = rcosθ and y = rsinθ. find the df(x,y)^2/d^2θ (second derivative with respect the theta) Homework Equations The Attempt at a Solution Don't exactly know what I'm doing here.. The notes from class give me this...

Under what conditions on the constants a and b does the second derivative test guarantee that the function g(x,y,z)=ax^2+2axz+by^2-2byz+z^2 has a local maximum at (0,0,0)? a local minimum at (0,0,0)? well, i used the Hessian matrix to compute the eigenvalues to set them above zero. but...

The curve C is defined parametrically by x = 4t - t^2 and y = 1 - e^-t where 0≤t<2 . Show that the mean value of d^2 y / dx^2 with respect to x over the interval 0≤x≤7/4 is (4e^(-1/2) - 3)/ 21 . I've figured out d^2 y/dx^2 as ((t-1)e^-t)/(4(2-t)^3) . Any idea how to do the the other part ?

Homework Statement I have the derived function: f'(x) = [1/(1+kx)^2]e^[x/(1+kx)] k is a positive constant Homework Equations I need to find the second derivative, which I thought was just the derivative of the exponent multiplied by the coefficient (as you find the...

Hi all! I was wondering if \partial_1\partial_2f=\partial_2\partial_1f in a Riemannian manifold (Schwartz's - or Clairaut's - theorem). Example: consider a metric given by the line element ds^2=-dt^2+\ell_1^2dx^2+\ell_2^2dy^2+\ell_3^2dz^2 can we assume that...

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial coordinates) R_{abcd}=\frac{1}{2} (g_{ad,bc}+g_{bc,ad}-g_{bd,ac}-g_{ac,bd}) where ",_i"...

1. Find the second derivative of the following function 2. f(x) = x\sqrt{5-x}, f'(x) = \frac{10-3x}{2\sqrt{5-x}} 3. f"(x)=\frac{-3*2\sqrt{5-x}-(10-3x)(2\sqrt{5-x})*-x}{(2\sqrt{5-x})^{2}} Here is where I get lost.

Okay! Earlier today I was thinking about potential energy and how it is related to an orbiting object, O, around a centre, C, from which force emanates if the object O is traveling at radius r from this centre, we conclude that the force given by the change in direction must be equal to the...

I'm having trouble thinking about the second derivative. I've been thinking of it as the rate of change of the rate of change, but that seems to have gotten me into some trouble. This is a quiz question that I had: http://i.imgur.com/WUMqY5C.jpg Ignore the first part, as it should read...

Homework Statement I'm trying to solve a system of two second order linear differential equations with the ode45 function. It is a two degree of freedom problem with 2nd order derivatives of both variables, u and theta. I believe that's referred to as a "stiff matrix"). I'm very...

The problem and my attempt at a solution is shown in the attached image. The problem is that I end up with one extra x in the denominator. So the question is: Is my expression for y'' correct and I just made a mistake somewhere (I have checked it several times), or am I missing something in the...

Hello all. In short, I am wondering what the second derivative of the Heaviside function (let's say H[(0)]) would be. I'm presuming that it's undefined (or more accurately, zero everywhere but at x=0), but I would like to know if that is correct. Essentially, I am attempting to extend a...

Homework Statement f(x) = √x^2 + x + 1Homework Equations Chain Rule The Attempt at a Solution f ' (x) = (x^2+x+1)^1/2 = 1/2 (x^2+x+1)^-1/2(x^2+x+1)' = 1/2(x^2+x+1)^-1/2(2x + 1) = 1/2(2x+1)/√(x^2+x+1) f '' (x) = So I am having trouble with that. Unless my answer for...

Homework Statement Determine the second derivative of y with respect to x when 2x2+3y2=0 Homework Equations possible answers include: 2/(3y2) -2/(9y3) 2/(3y3) -2/(3y2) -2/(3y3) The Attempt at a Solution I took the first derivative with respect to x implicitly and came up with...

1. y = (sinx)^2 Homework Equations The Attempt at a Solution i found the 1st deriviative... 2(sinx)(cosx) and then the second derivative... (-2(sinx)^2) + (cosx)^2 i don't have the answer. so i wanted to know if this was right.

I posted this in two forums because the question might be too chemistry related for the people in the calculus forum. I'm a visual thinker so I struggle a bit to get my head around calculus concepts. So as an example, here's a potential energy surface: lets say this represents the structure of...

I'm a visual thinker so I struggle a bit to get my head around calculus concepts. So as an example, here's a potential energy surface: lets say this represents the structure of a simple molecule like n-propane: the molecule in the picture is the most stable conformation so the global minimum...

Hi guys, I have this function f(g(t)) and I have to find the second time derivative of f, is it correct the following solution?: f''=∂f/∂g*g'=∇f*g' f ''=∇^2f*|g'|^2+∇f*g'' where ∇^2 is the laplacian function

I'm using variable substitution to solve a problem. Finding the relationships between the first derivatives of both sets is straightforward using the chain rule, but I'm uncertain if the way I'm determining the second derivative relationships is correct. Given a description of a problem...

Homework Statement Find the second derivative of 3(x^2)y+y+x=x^5 Homework Equations Find the first derivative using implicit differentiation. Find the second by using quotient rule. The Attempt at a Solution So I found the first derivative to be dy/dx = 5(x^4)-6xy-1 / 3(x^2)+1...

The second derivative at a maximum is either negative or zero. Can you explain how it can be zero? There can't be a 'plateau' at the maximum or it would not be a point. I cannot imagine graphically how the second derivative at a maximum can be zero. Before the maximum, the gradient is...

Homework Statement So fx is how much f changes when you change x. Thus fxx is the rate of change of fx, or geometrically how fast the functions slope is changing. The same can be said for fy and fyy. But what about fxy and fyx? Could someone please explain to me what they mean? I want...

Homework Statement I'd always used the 2nd derivative test for the nature of stationary points. But I came across this equation in one of my exercises, for which the test doesn't seem to work at all. Find the stationary points of: y=(x^2-1)4, stating the nature of each. Homework...

I'm not sure how to evaluate the second derivative d2y/dx2 = (4(dy/dx) -12x^2 -12y^2(dy/dx)^2)) /(4y^3 - 2x) AT A GIVEN POINT (1,1). The answer is -14 not sure how they got it. Final exam this friday would appreciat e answers very much!

Homework Statement Attached Homework Equations The Attempt at a Solution At (-1, -5), moving up along y causes a drop in the value of the function, so the first derivative wrt y is negative. Since the contour lines also get closer in this direction, the function is dropping...

http://www.math.northwestern.edu/courses/placement/220_Self_Placement.pdf Question 7 here involves a function with the rule f(x) = 3x5-5x3. I computed the 2nd derivative as f''(x) = 60x3-30x (Mathematica agrees.), giving inflection points for f at -1/sqrt(2), 0, 1/sqrt(2). But the answer...

Question: How can I plot the derivative of a function? Following code for problem 5.9 in MatLab by Gilat 4th Ed: t=[0:1:20]; theta=pi*(1-exp(-0.2*t)); r=20+30*(1-exp(-0.1*t)); v=diff(theta) table=[t' theta' r'] polar(theta,r) polar(theta,v) Error message: ? Error using ==>...

Homework Statement Derive an expression for the composition of 2 functions. Homework Equations The Attempt at a Solution I started with supposing that y(x) = f(g(x)). I know that dy/dx = df/dg * dg/dx (via the chain rule). Doing the derivative again, I started with the product...

Homework Statement Solve the following equation for x(y). (use no differential functions) x(0)' and x(0) are known.Homework Equations The Attempt at a Solution I'm a bit unsure as to what to do next but I can easily make a messy formula up to approximate the result. k1 and k2 are a...

Hi all... I've read on wikipedia (facepalm) that the first derivative of a determinant is del(det(A))/del(A_ij) = det(A)*(inv(A))_j,i If we go to find the second derivative (applying power rule), we get: del^2(A) / (del(A)_pq) (del (A)_ij) = {del(det(A))/del(A_pq)}*(inv(A))_j,i...

I've been trying to use Taylor's theorem with h = (y-x)/2 to show that a twice differentiable function for which the second derivative is positive is midpoint convex (ie, f( (1/2)*(x+y) ) \leq (1/2) * (f(x)+f(y)) ). (It's not a homework problem.) The problem I end up with this is that I'm not...

Hi there, just wanted to make a clarification before my final exam. The second derivative test for partial derivatives (or at least part of it) states if D = ∂2f/∂x2 * ∂2f/∂y2 - (∂2f/∂x∂y)2 and (a,b) is a critical point of f, then a) if D(a,b) > 0 and ∂2f/∂x2 < 0, then there is a local...

Homework Statement Find y" in terms of x and y: y^2 + 2y = 2x + 1 Homework Equations N/A The Attempt at a Solution I found the first derivative: y^2 + 2y = 2x + 1 2yy'+2y'=2 2y'.(y+1)=2 y'=2/2(y+1) y'=1/(y+1) But I'm having trouble moving on from there.