What is Second derivative: Definition and 178 Discussions
In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation:
a
=
d
v
d
t
=
d
2
x
d
t
2
,
{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}},}
where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change. The last expression
d
2
x
d
t
2
{\displaystyle {\tfrac {d^{2}{\boldsymbol {x}}}{dt^{2}}}}
is the second derivative of position (x) with respect to time.
On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.
(i) I take the second derivative of Y: Y'' = 6X + 2A. Y'' = 0 when X = -A/3. Moreover, as Y'' is linear it changes sign at this X. Thus, it is the point of inflection.
(iii) After the substitution, the term x^2 appears twice: one, from X^3 as -3(x^2)(A/3), and another from AX^2 as Ax^2. They...
Find text (question and working to solution here ...this is very clear to me...on the use of implicit differentiation and quotient rule to solution). I am seeking an alternative approach.
Now from my study we can also have; using partial derivatives...
Let's say we have a function ##M(f(x))## where ##M: \mathbb{R}^2 \to \mathbb{R}^2## is a multivariable linear function, and ##f: \mathbb{R} \to \mathbb{R}^2## is a single variable function. Now I'm getting confused with evaluating the following second derivative of the function:
$$
[M(f(x))]''...
ok this is pretty straightforward to me, my question is on the order of differentiation, i know that:
##\frac {d^2y}{dx^2}=####\frac {d}{dt}.####\frac {dy}{dx}.####\frac {dt}{dx}##
is it correct to have,
##\frac {d^2y}{dx^2}=####\frac {d}{dt}##.##\frac {dt}{dx}##.##\frac {dy}{dx}##?
that is...
Q: See f(t) in graph below. Does the graph of g have a point of inflection at x=4?
There is a corner at x=4, so I don't think there is a point of inflection. Does a point of inflection exist where f''(x) does not exist? The solution says there is a point of inflection, could anyone explain why...
If the sign on the sign diagram of f" changes from positive to negative or from negative to positive, this means the critical points of f" is non-horizontal inflection of f
But what about if the sign does not change? Let say f"(x) = 0 when ##x = a## and from sign diagram of f", the sign on the...
Since distances increase, their first derivative which is velocity (Hubble constant) should be positive if not increasing too. Accelerated expansion needs the velocity to increase. What about the third derivative which is acceleration? An accelerated universe could have third derivative (called...
Hello. My understanding of the importance of second derivatives is that they help us to know whether the graph of a function is concave upward or concave downward. In the equation ## f(x) = x^2 + 2x ## we already know from the first derivative, ## f\prime (x) = 2x + 2 ##, that the graph is...
If $(x+2y)\cdot \dfrac{dy}{dx}=2x-y$ what is the value of $\dfrac{d^2y}{dx^2}$ at the point (3,0)?
ok not sure of the next step but
$\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$
In one of my textbooks about quantum mechanics, they mention a vehicle moving in a straight line along the x axis. With Newtons first law they take the second derivative from a which is
d^2x/dt^2 and that should be equal to
-∂V/∂x. What exactly does -∂V indicate?
The complete equation...
Hello all.
I was playing around with the time dilation equation : √(1-v2/c2)
Specifically, I decided to take the derivative(d/dv) of the equation. Following the rules of calculus, as little of them as I know, I got this:
d/dv(√(1-v2/c2) = v / (c2√(1-v2/c2)).
Now, this seems reasonable enough...
Homework Statement
This is a translation so sorry in advance if there are funky words in here[/B]
f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ.
Show that the sequence on functions $$ n[f(x + 1/n) - f(x)] $$ converges uniformly on f'(x) on ℝ...
In my classical mechanics course, the professor did a bit of algebraic wizardry in a derivation for one of Kepler's Laws where a second derivative was simplified to a first derivative by taking the square root of both sides of the relation. It basically went something like this:
\frac{d^2...
Homework Statement
Let ##f: \mathbb{R} \rightarrow \mathbb{R}## a function two times differentiable and ##g: \mathbb{R} \rightarrow \mathbb{R}## given by ##g(x) = f(x + 2 \cos(3x))##.
(a) Determine g''(x).
(b) If f'(2) = 1 and f''(2) = 8, compute g''(0).
Homework Equations
I'm not aware of...
Hi all
I am trying to reproduce some results from a paper, but I'm not sure how to proceed. I have the following: ##\phi## is a complex matrix and can be decomposed into real and imaginary parts:
$$\phi=\frac{\phi_R +i\phi_I}{\sqrt{2}}$$
so that
$$\phi^\dagger\phi=\frac{\phi_R^2 +\phi_I^2}{2}$$...
Homework Statement
Only the second part
Homework Equations
Second derivative:
$$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}$$
The Attempt at a Solution
$$dx=(1-2t)\,dt,~~dy=(1-3t^2)\,dt$$
Do i differentiate the differential dt?
$$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$...
Homework Statement
In textbook i was given formula to calculate error.
I know that:
E(t) = f(t)- L(x) = f(t) - f(a)- f'(a)(t- a) [L(x) is linear approximation]; [Lets call this Formula 1]
I understand that, but that I have formula:
E(x) = f''(s)/2 * (x-a)^2 [lets call this Formula 2]
Here...
Firstly I know how to do this with first derivatives in differential equations - for example say we had ##\frac{dy}{dx}=4y^2-y##, and we're also told that ##y=1## when ##x=0##.
##\frac{dy}{dx}=4y^2-y##
##\frac{dx}{dy}=\frac{1}{4y^2-y}=\frac{1}{y\left(4y-1\right)}=\frac{4}{4y-1}-\frac{1}{y}##...
I'm studying boundary layers. I am confused by what I am reading in this book.
The book says the friction force (F) per unit volume = $$\frac{dF}{dy}=\mu\frac{d^2U}{dy^2}$$
They say $$\frac{dU}{dy}=\frac{U_\infty}{\delta}$$
This makes sense to me, delta is the thickness in the y direction...
Hey! :o
I want to find the first and second derivative of the function $$\psi (\lambda )=f(\lambda x_1, \lambda x_2)$$ where $f(y_1, y_2)$ is twice differentiable and $(x_1, x_2)$ is arbitrary for fix.
I have done the following:
$$f(g(\lambda), h(\lambda)) : \\...
Homework Statement
Question has been attached to topic.
Homework Equations
Chain rule.
The Attempt at a Solution
$$\frac {dy}{dt} . \frac{dt}{dx} = \sqrt{t^2+1}.cos(π.t)$$
$$\frac{d^2y}{dt^2}.(\frac{dt}{dx})^2 = 2 $$
$$\frac{d^2y}{dt^2}.(t^2+1).cos^2(π.t)= 2 $$ and for the t=3/4...
Quick question. I know that if we have a curve defined by ##x=f(t)## and ##y=g(t)##, then the slope of the tangent line is ##\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}##. I am trying to find the second derivative, which would be ##\displaystyle \frac{d}{dx}\frac{dy}{dx} =...
Homework Statement
Develop aprogram that will determine the second derivative of pi(16 x^2 - y^4) at y=2 with step sizes of 0.1, 0.01, 0.001…. until the absolute error (numerical-analytical) converges to 0.00001. Use the 2nd order Central Difference Formula.
User Input: y, tolerance
Output: h...
I am trying to numerically solve a PDE, and just had a question as to the validity of a certain approach. For example, given the PDE:
$$ \frac {\partial ^2 E}{\partial t^2} = - k\frac {\partial E}{\partial t} + c^2 \frac {\partial ^2 E}{\partial z^2} - c\frac {\partial E}{\partial z}$$
If I...
Homework Statement
Find y''
Homework Equations
9x^2 +y^2 = 9
The Attempt at a Solution
y'
18x+2y(y')=0
y'=-18x/2y
y'=9x/y
For the second derivative, I get the correct answer (same as the book) up until the very last step.
Here's where I'm left at:
-9( (-9x^2 - y^2) / y^3 )
The book then...
I have a set of data as follows, How can I calculate the second derivative of the curve obtained from these data.
x=[0.1;0.07;0.05;0.03;0]; r=[-98.9407;-105.7183;-111.2423;-116.0320;-120.0462];
The formula T = -(ħ/2m)∇2 implies that T is proportional to the second spatial derivative of a wavefunction. What is the origin of this dependence?
In classical mechanics, T = p2/2m. Is it also the case in classical mechanics that p2/2m is proportional to a second spatial derivative? I...
So to find the x values of the stationary points on the curve:
f(x)=x3+3x2
you make f '(x)=0
so:
3x2+6x=0
x=0 or x=-2
Then to find which of these points are maximum or minimum you do f ''(0) and f ''(-2)
so:
6(0)+6=6
6(-2)+6=-6
so the maximum has an x value of -2 and the minimum has an x value...
Hello. I have a question regarding curvature and second derivatives. I have always been confused regarding what is concave/convex and what corresponds to negative/positive curvature, negative/positive second derivative.
If we consider the profile shown in the following picture...
Hi,
I've attached an image of an equation I came across, and the text describes this as an approximation to the second derivative. Everything seems to be exact to me (i.e. not an approximation) if the limit of h was taken to 0. Is that the only reason why it's said to be an approximation or is...
Wolfram and the Linear Algebra text I'm currently working on, give the two possible solutions of \frac{d^2y}{dx^2}=y as being e^{x} and e^{-x}, or rather, constant multiples of them.
Here wolfram agrees:
http://www.wolframalpha.com/input/?i=d^2y/dx^2=y
My question is, why isn't y = e^{x} + x...
Hi,
I feel sometimes when I'm doing calculus I lose the logic and intuition behind what I'm doing, especially when integrating. I have yet to find a way to think about it in a way it makes sense to me why the definite integral would tell us the area under a curve. Same with why the second...
Homework Statement
The displacement of a machine is given by the simple harmonic motion as x(t) = 5cos(30t)+4sin(30t). Find the amplitude of motion, and the amplitude of the velocity.
Homework Equations
x''(t) = -4500cos(30t)-3600sin(30t)
The Attempt at a Solution
[/B]
I should note that...
http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx
On this page the author makes it very clear that:
$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$
provided ##\frac{dx}{dt} \neq 0##.
In example 4, ##\frac{dx}{dt} = -2t##, which is zero when ##t## is zero. In simplifying...
Using the standard equation of a circle x^2 + y^2 = r^2, I took the first and second derivatives and obtained -x/y and -r^2/y^3 , respectively. I understand that the slope is going to be different at each point along the circle, but what does not make sense to me is that the rate of change of...
According to this link: http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx
The second derivative test can only be applied if ##f''## is continuous in a region around ##c##.
But according to this link...
SOLVED
1. Homework Statement
Find polynomial of least degree satisfying:
p(1)=-1, p'(1)=2, p''(1)=0, p(2)=1, p'(2)=-2
Homework Equations
In general, a Hermite Polynomial is defined by the following:
∑[f(xi)*hi(x)+f'(xi)*h2i(x)]
where:
hi(xj)=1 if i=j and 0 otherwise. Similarly with h'2...
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..
Hi,
I'm trying to find the eigenvalues and eigenvectors of the operator ##\hat{O}=\frac{d^2}{d\phi^2}##
Where ##\phi## is the angular coordinate in polar coordinates.
Since we are dealing with polar coordinates, we also have the condition (on the eigenfunctions) that ##f(\phi)=f(\phi+2\pi)##...
What does it mean when I have to find the second derivative of a circle at a given point? (Implicit diffing)
In specifics, the equation is 9x2 +y2 =9
At the point (0,3)
You don't really need the rest at all, but it was just my process.
This seems to make no sense.
first D'v 18x+2yy'=0
Second...
For the derivative: dy/dt = ry ln(K/y)
I am trying to solve the second derivative. It seems like an easy solution, and I did:
d^2y/dt^2 = rln(K/y)y' + ry(y/K)
which simplifies to:
d^2y/dt^2 = (ry')[ln(K/y) + 1/Kln(K/y)
Unfortunately, the answer is d^2y/dt^2 (ry')[ln(K/y) - 1] and I don't...
In the Feynman Lectures on Physics chapter 28, Feynman explains the radiation equation $$\vec{E}=\frac{-q}{4\pi\epsilon_0 c^2}\,
\frac{d^2\hat{e}_{r'}}{dt^2}$$
The fact that the transverse component varies as ##\frac{1}{r}## seems fairly obvious to me since what matters is just the angle...
I am doing critical points and using the second derivative test (multivariable version)
Homework Statement
f(x,y) = (x^2+y^2)e^{x^2-y^2}
Issue I am having is with the system of equations to get the critical points from partial wrt x, wrt y
The Attempt at a Solution
f_{x} =...