Calculus Definition and 1000 Threads

I have to find functions that maximise certain criterea. The problem can however not be put "under a single integral", for example I've to find ##f(t)##, ##g(t)## that maximise: ## \int_0^{t_e}f(t)^2dt\int_0^{t_e}g(t)^2dt - (\int_0^{t_e}f(t)g(t)dt)^2 ## With ## -1 \leq f(t)\leq1## and ## -1...

Homework Statement Can someone please check my working, as I am new to Einstein notation: Calculate $$\partial^\mu x^2.$$ Homework Equations 3. The Attempt at a Solution [/B] \begin{align*} \partial^\mu x^2 &= \partial^\mu(x_\nu x^\nu) \\ &= x^a\partial^\mu x_a + x_b\partial^\mu x^b \ \...

Homework Statement Give the example and show your understanding: [1][/B].Lets define some property of a point of the function: 1. Point is a stationary point 2. Point is a max/min of a function 3. Point is a turning point of a function If possible name a function whose point has properties of...

I have taken precalculus class a few years ago. I finally took calculus 1 and the same school just last quarter and got a 3.2. I had a pretty cool teacher. I'm transferring to a four year school and have to retake calculus 1, but I'm contemplating on retaking a pre calculus at the four year...

Homework Statement ##\frac{e^x-1}{x}## Evaluate the limit of the expression as x approaches 0. Homework Equations 3. The Attempt at a Solution [/B] The question i have is more theoretical. I was able to solve this problem by expanding the expression into the talyor polynomial at ##x=0##. I...

Homework Statement "Derive the equations for position (in terms of acceleration, initial position, initial velocity, and time) and velocity (in terms of constant acceleration, a, initial velocity, v0, and time, t) from the definitions of position, velocity, and acceleration (derivative...

Homework Statement This is one test question we had today and it asks as to provide examples of functions and intervals. Some may be untrue so we had to identify it. The test isn't graded yet so these are my question answers. Hopefully you'll correct me where necessary and provide a true...

So I am currently a very indecisive mechanical engineering student, who can't figure out what to major in. I have found out that I am much more interested in solving problems that deal with a lot of equations, substitution, and differential equations than I am solving statics problems. I like...

Homework Statement Compute the work of the vector field ##F(x,y)=(\frac{y}{x^2+y^2},\frac{-x}{x^2+y^2})## in the line segment that goes from (0,1) to (1,0). Homework Equations 3. The Attempt at a Solution [/B] My attempt (please let me know if there is an easier way to do this) I applied...

I have recently been attempting to solve a problem that has been bugging me for quite some time. I've gotten back into calculus and integrals to attempt to solve a little formula I'm trying to build for a simulation test. Over-all, if I have ##\int _0^bv^2x\ dx## I'd expect the outcome to be...

I found a deduction to determinate de sum of the first n squares. However there is a part on it that i didn't understood. We use the next definition: (k+1)^3 - k^3 = 3k^2 + 3k +1, then we define k= 1, ... , n and then we sum... (n+1)^3 -1 = 3\sum_{k=0}^{n}k^{2} +3\sum_{k=0}^{n}k+ n The...

Dear all, I'm currently teaching calculus courses with Stewart's book. I was wondering what other teachers their experiences are with giving fully worked out answers to the exercises versus giving just hints and the final answer. I have the feeling that the last approach activates students...

Homework Statement Sketch the curve with the given vector equation. Indicate with an arrow the direction in with 't' increases r(t)=<t, 2-t, 2t>Homework Equations parametric equation (can't type the equation, too confusing to use the template) The Attempt at a Solution So far, I have <1, -1...

Hi guys, This is my first time posting on PF! I have a question on §3 of Einstein's paper "On the electrodynamics of moving bodies."My problem is with the following mathematical statements: Hence, if x' is chosen to be infinitesimally small, or I have just finished high school, and...

Homework Statement For which value of d does the following limit exist? lim x->d ln [ (x2-13x+30) / (x-d) ] Homework Equations None The Attempt at a Solution I understand how to find limits when the limit goes to a real number, and has a variable in the function to solve for, but not when...

Homework Statement If ## \lim_{x \rightarrow a} f(x) = \infty## and ##\lim_{x \rightarrow a} g(x) = c ##, and if ##c>0## then prove that ##\lim_{x \rightarrow a} \left[ f(x)g(x)\right] = \infty~~ \text{if c > 0}## Homework Equations Epsilon Delta definition of the limit The Attempt at a...

Homework Statement Show that Dx∫f(u)du = f(x) Where the integral is evaluated from a to x. (Hint: Do Taylor expansion of f(u) around x). Homework Equations None The Attempt at a Solution I have ... = Dx(F(u)+C) = Dx(F(x-a)+C) = dxF(x) - dxF(a) = f(x)-f(a). My problem is that it should be...

Homework Statement find equation of plane P that is perpendicular to line L which passes through the point (-2,-2,3) Homework Equations ... The Attempt at a Solution [/B]line L passes through the points (1,2,1) and (0,0,-3) I have worked out the parametric equations of line L to be x=1-t...

Homework Statement Calculate the integral: ## \int_{a}^{b} \frac{1}{x} dx ## Homework Equations - The Attempt at a Solution In high school we learned that: ## \int_{a}^{b} \frac{1}{x} dx = ln(|x|) + C ## because the logarithm of a negative number is undefined. However, in my current maths...

I'm a 10th grade student in the United States and currently taking geometry which is a breeze, and if anyone else reading this is in the U.S. you know that 10th graders haven't reached calculus yet, not even physics. Since i know I'm going into quantum physics, i have thirst to learn calculus...

Evaluate the integral \iiint\limits_{ydV}, where V is the solid lying below the plane x+y+z =8 and above the region in the x-y plane bounded by the curves y=1, x=0 and x=\sqrt{y}.

Homework Statement Let f(x, y) = x^2 + kxy + y^2 , where k is some constant in R. i. Show that f has a stationary point at (0, 0) for every k ∈ R Homework Equations ... The Attempt at a Solution I may have the solution or i may have gone completely wrong I am not entirely sure. i first found...

Homework Statement Consider the function $$ f(x) := \begin{cases} 0, & if~ x \in (0,1) - \mathbb{Q} \\ \frac{1}{q}, & if~ x = \frac{p}{q} \in (0,1) \cap \mathbb{Q} \mbox{ in lowest terms. } \\ \end{cases} $$ In his Calculus book, Spivak shows us that ##\lim_{x \rightarrow a} f(x) = 0## for...

Hi, this is a newbee question. Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))? That is, the two-dimensional area under a curve in [a,b] for f(x) is always equals to the one-dimensional distance F(b)-F(a)? If...

Homework Statement We're given the gaussian distribution: $$\rho(x) = Ae^{-\lambda(x-a)^2}$$ where A, a, and ##\lambda## are positive real constants. We use the normalization condition $$\int_{-\infty}^{\infty} Ae^{-\lambda(x-a)^2} \,dx = 1$$ to find: $$A = \sqrt \frac \lambda \pi$$ What I want...

Prove (formall y) in predicate calculus : $\neg\exists y\,\forall x\,(x\in y\leftrightarrow \neg x\in x)$.

Homework Statement f(x,y) = 1/y^2-x find the domain of f. Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f. Homework Equations I know that the domain of the function is anywhere that the function is defined. The Attempt at a Solution in the case of...

Basically i need to evaluate the limit of this expression ##\lim \sqrt[3]{n^6-n^4+5}-n^2=?## I want to know if this is correct and why: ##\lim \sqrt[3]{n^6-n^4+5}-n^2=\lim...

Finding length of vector with unknown variable. Purely for study purposes. Find the smallest possible length of the vector →v . Let vector V = (-2/3,b,16/7) Equation for finding length of vector : Sqrt(a^2+b^2+c^2)Question would be quite straight forward had there been no unknown variable but...

Homework Statement We are given a Banana, and asked to find the volume and surface area of the function, using calculus. So far, we have learned elementary calculus (derivatives, limits, and integrals) as well as volumes of revolutions. We traced the banana on graph paper, plotted points on the...

So, I am majoring in mechanical engineering with a minor in physics at San Jose State University. I want to learn as much as possible with physics by reading books and taking my future courses at SJSU, but I don't know what books to read. Any recommendations? I want to get pretty close to...

Okay, so I'm currently a mechanical engineering major at San Jose State University, but I just want to become much more engaged with physics and mathematics. I do pretty good with calculus, math comes easy to me. I'm a first year student taking calculus 2. I was just wondering what books I can...

Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is $$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...

Homework Statement If f(x) is a scalar-valued function, show that ∂ƒ²/∂xi∂xj are the components of a Cartesian tensor of rank 2. Homework Equations N/A The Attempt at a Solution I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't think this is...

Homework Statement Homework Equations The Attempt at a Solution I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus. This is my attempt at the cross product and...

Homework Statement find the anti-derivative of xe ^x so its x to the power of e to the power of x Homework EquationsThe Attempt at a Solution i have 0 idea where to even start. this was a question on my quiz today

Homework Statement -here is the problem statement -here is a bit of their answer Homework Equations Chain rule, partial derivative in spherical coord. The Attempt at a Solution I tried dragging out the constant and partial derivate with respect to t but still I can't reach their df/dt and...

Homework Statement ∞ Σ (-1)n-1 n/n2 +4 n=1 Homework Equations lim |an+1/an| = L n→∞ bn+1≤bn lim bn = 0 n→∞ The Attempt at a Solution So I tried multiple things while attempting this solution and got inconsistent answers so I am thoroughly confused. My work is on the attached photo. I found that...

I'm tutoring a student who is in a typical precalculus/trig course where they're teaching her about graphing various arbitrary polynomials. Among the rules of multiplicity and intercepts they seem to be phrasing the questions such that they expect the students to also find the maxima and minima...

In our section on path independence, we were asked to find the potential function given a vector field. Our teacher says to use only line integrals to find the potential function, and not any other method. Like if we have ##F=\left< M,N,P \right> ## The first step is to determine if the domain...

Hello everyone. I'm currently a sophomore working toward a BS in physics (and a minor in astronomy) at a top private engineering school. With the semester finishing up, I'm a little worried about where I am now and where I will be after graduation and I have a few questions. A little...

Homework Statement A marine biology observation pod is being designed. It will be submerged, with vertical windows of various sizes and shapes. Calculate the total force being exerted on each of the windows described below. Density of water is 1000 kg/m^3...

Hi everybody, I’m trying to calculate the shape of a boundary line f(x) between two mediums that collimates rays from a point light source. This requires the rays to hit the boundary line under a certain angle, so I calculated the slope m(φ) of the boundary line for a ray with polar angle φ (φ...

Homework Statement A mass m is suspended by a light elastic string. When the mass remains at rest it is at a point 0, which is a distance a + b below the point from which the string is suspended from the ceiling, where a is the natural length of the string. The mass is pulled down a distance h...

Hello. I finished working through Spivak's Calculus 3rd edition chapters 13 "Integrals", and 14 "The Fundamental Theorem of Calculus". By that I mean that I read the chapters, actively tried to prove every lemma, theorem and corollary before looking at Spivak's proofs, took notes into my...

Good Morning Could someone please distinguish between the Frechet and Gateaux Derivatives and why one is better to use in the Calculus of Variations? In your response -- if you are so inclined -- please try to avoid the theoretical foundations of this distinction (as I can investigate that by...

Homework Statement We are given a disk with negligible thickness, a radius of 1m, and a surface charge density of σ(x,y) = 1 + cos(π√x2+y2). The disk is centered at the origin of the xy plane. We are also given the location of a point charge in Cartesian coordinates, for example [0.5,0.5,2]. We...

Homework Statement Sorry that I am not up on latex yet, but will describe the problem the best I can. On the interval of a=1 to b= 4 for X. ∫√5/√x. Homework EquationsThe Attempt at a Solution My text indicates the answer is 2√5. I have taken my anti derivative and plugged in b and subtracted...

A simple method to find the potential of a conservative vector field defined on a domain ##D## is to calculate the integral $$U(x,y,z)=\int_{\gamma} F \cdot ds$$ On a curve ##\gamma## that is made of segments parallel to the coordinate axes, that start from a chosen point ##(x_0,y_0,z_0)##. I...

If a brick is pulled across the floor by a rope thruogh a pulley, 1 meter above the ground - and work = W, where W = 10N , (in Newton).Show that the horizontal component of W, which is pulling the brick has the size \frac{10x}{\sqrt{1+x^2}} (*) Use this to calculate the amount of work needed...