Derivatives Definition and 1000 Threads

I had questions on 2 Problems in the Text: 1. The total cost C of producing x units of some item is a function of x. Economists use the term marginal cost for the rate of change of C with respect to x. Suppose that: C = 5x^2 + 15x + 200 What is the marginal cost when x = 15? Would this...

Homework Statement Use the polar graph to determine the signs (+,-,0) of each derivative at the point labeled A. Homework Equations dy/dx= dy/dtheta= dx/dtheta= dr/dtheta= The Attempt at a Solution Hi people, I need help with this question. See the picture of the graph...

I have a PDE in two variables, u and v, which takes the form \frac{\partial\psi}{\partial u\hspace{1pt}\partial v} + \frac{1}{r}\left(\frac{\partial r}{\partial u} \frac{\partial \psi}{\partial v} + \frac{\partial r}{\partial v}\frac{\partial\psi}{\partial u}\right) for an auxiliary...

Homework Statement f(x.y)=4x^2-y^2 Homework Equations Ʃ partial derivative components(?) The Attempt at a Solution The solution when θ=pi and f(1,-1) is -8. Does this mean that one of the coordinates of this function is (1,-1,-8)? What exactly is the directional derivative, and what does...

How can I use the directional derivative of a two variable function to show that the limit does not exist? For example, suppose I have a function f(x,y)=g(x)/f(y) and g(a)=f(b)=0 and the limit as x and y go to a and b is 0. How would I use the directional derivative to show that the limit at...

We have a function f:R^2->R and it has partial derivative of 2nd order. Show that f_{xy}=0 \forall (x,y)\in \mathbb{R}^2 \Leftrightarrow f(x,y)=g(x)+h(y) The <= is self explanatory, the => I am not sure I got the right reasoning. I mean we know that from the above we have: f_x=F(x) (it's...

Homework Statement The question says ti find the derivative of y with respect to the independent variable. The equation is: y=4 ln 2t. Homework Equations I know how to find derivatives using the product/quotient/chain/etc rules, but that isn't what the question is asking for. I...

Homework Statement hey Forum! I had a question here I'm struggling with and was wondering if someone could take a look. its Dealing with calculus, specifically derivatives and behaviors of the graph: http://i41.tinypic.com/mc6opj.jpg I just started and part a) already has me stumped D...

Homework Statement Hey forum, hope everyone is having a good day! If someone could check this question out for me that'd be great! Determine the first and second derivative: g(x) = (2x - 3)/(x + 4) The Attempt at a Solution g(x) = (2x – 3)(x + 4)^(-1) g’(x) = (2x – 3)’(x +...

Hi all, I've been fiddling around with this problem for a while. I intuitively understand that the parallel propagator is the path integral of the connection. I would like to be able to show the converse (connection is derivative of parallel propagator) mathematically, and I am having a...

Homework Statement Find the partial derivatives: f(x,y)= integral[x,y] cos(t^2)dt, find f_x(x,y) and f_y(x,y) Homework Equations I know from calculus that the derivative of an integral is the function. The Attempt at a Solution I found that the integral of [x to y]...

Homework Statement Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g differentiable. (a) What are the values of the derivatives f'(1), g'(1)? (b)...

Homework Statement a) d^2/(dx^2): ln(x+1) b) d^3/(dx^3): x^7 + 4x^6 - x^2 c) d^2/(dx^2): 1/(x + 1) d) d^4/(dx^4): cos(2x) Homework Equations none The Attempt at a Solution Can someone tell me if these are right? a)= -(1/(x + 1)^2) b)=210x^4 + 480x^3 c)= (2/(x + 1)^3) d)=...

Homework Statement The question is attached along with its solution. Homework Equations Partial differentiation and the implicit function theorem. The Attempt at a Solution My work is attached. I feel it's correct but is it incomplete? I have the following questions/confusions...

How do you solve (analytically or numerically) a differential equation of this form, $$\frac{\mathrm{d}y(x)/\mathrm{d}x}{\mathrm{d}z(x)/\mathrm{d}x} = a[1-y(x)-z(x)] + b$$ where a, b are constants. Also, $$y(0) = z(0) = 0$$

Homework Statement Assume f(1,1,1)=3 and f(1.1,1.2,1.1)=3.1 a) Which directional derivative Duf at (1,1,1) can be estimated from this information? Give vector u b) Estimate the directional derivative in part a Homework Equations Duf = del f (dot product) vector u del f =...

Homework Statement My question is how do I take the time derivative of (theta dot)^2? Homework Equations The Attempt at a Solution Is the answer just 2(theta double dot)^1 or do you use chain rule 2(theta dot)*(theta double dot)?

Homework Statement f(x)=X^2/(x^2-16) f(x)=1+x/1-X f(x)=X^3(X-2)^2 Ive done the first and second derivatives but they just don't seem right Homework Equations Quotient/Chain/Product Rule The Attempt at a Solution f(x)=X^2/(x^2-16) (X^2-16)(2X)-(X^2)(2X)/(X^2-16)^2...

Homework Statement Just trying to find the first and second derivatives. X^2/(X^2-16) 1+X/1-X X^3(X-2)^2 Homework Equations Quotient Rule/Power Rule/Chain Rule The Attempt at a Solution

Homework Statement Do the derivatives del and d/dt commute? Or in other words, is it true that: del(d/dt)X = (d/dt)del_X Homework Equations ? The Attempt at a Solution nm, I think I know how to show it now..

Homework Statement In the steps below, the ∂z/∂x does not seem to be obeying normal algebraic rules. I'm confused. This is not really a problem, I'm just trying to understand the steps. The Attempt at a Solution 1. 3z2∂z/∂x - y + y∂z/∂x = 0 2. ∂z/∂x = y/(y + 3z2) if ∂z/∂x were...

If you have a function f(x,y)=xy where y is a function of x, say y=x^2 then the partial derivative of f with respect to x is \frac{\partial f}{\partial x}=y However, if you substitute in y and express f as f(x)=x^3 then the partial derivative is \frac{\partial...

Homework Statement find the direction of P sub 0 in the direction of A see second post for attachment, I forgot to place it on this one. The Attempt at a Solution I'm only worried about the part that says gy(x,y,z) = -3zexsin yz. I also don't understand the conversion of gx and gz. 1. why...

Homework Statement ∂f/∂x (xy -1)2 = 2y(xy-1) The Attempt at a Solution I would think the answer would be 2(xy-1) I don't understand where the y comes from in 2y

Homework Statement I don't understand why ∂f/∂x = xy = y whereas ∂f/∂x = x2 + y2 = 2x Why does the y disappear in the second but not in the first?

Homework Statement Given that the acceleration vector is a(t) = (-16cos(-4t))i + (-16sin(-4t))j + -2tk, the initial velocity is v(0) = i + k and the initial position vector is r(0) = i + j + k, compute: The velocity vector v(t) = ___i + ____j + ____k The position vector r(t) = ___i +...

Are the following equalities between total and partial derivatives true if \frac{dy}{dx}=f(x,y)? \displaystyle \frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} f(x,y) \displaystyle \frac{d^2f}{dx^2} = \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} +...

Homework Statement Classify the equation and use the change of variables to change the equation to the form with no mixed second order derivative. u_{xx}+6u_{xy}+5u{yy}-4u{x}+2u=0 Homework Equations I know that it's of the hyperbollic form by equation a_{12}^2 - a_{11}*a_{22}, which...

Homework Statement Suppose the function f:R^2→R has 1st order partial derivatives and that δf(x,y)/δx = δf(x,y)/δy = 0 for all (x,y) in R^2. Prove that f is constant; there exists c such that f(x,y) = c for all (x,y) in R. There's a hint as well: First show that the restriction of...

How do CAS systems and programmable calculators evaluate the derivative of a function? Do they use matrix representation of linear transformations?

I was reading a section on vector fields and realized I am confused about how to take partials of vector quantities. If V(x,y)= yi -xj, I don't understand why the \partialx= y and the \partialy= -x. The problem is showing why the previous equation is not a gradient vector field (because the...

Can anyone give me an example of a continuous function that is NOT differentiable(other than the square root function)? I have to prove that not all continuous functions are differentiable. Thanks!

If a and b are constants, compute the expression KY'(K) + LY'(L) for Y = AK^a + BL^a Y'(K) means partial derivative with respect to K by the way. The answer in the book is KY'(K) + LY'(L) = aY I'm not sure what they did or what they're asking :/

Homework Statement I want to show that the partials exist for a certain function. Homework Equations My book says that if a function f is differentiable at a point x then the partial derivatives exist. The Attempt at a Solution Rather than showing f is differentiable, I am...

I just ran into this problem and have no idea how to solve it. Basically I'm trying to prove that all orders of derivative of the given function is bounded by the function on the right. I'm pretty sure the inequality is true, but I really have no clue on how to prove it. I thought about using...

In this textbook, how exactly are 2.2.19 and 2.3.21 equal? http://i.imgur.com/LrcXE.png

Homework Statement In one of my class's tests I've come across the following equation: \frac{d^2 y}{dx^2} \: + \: \left(\frac{dy}{dx}\right)^3 \frac{d^2 x}{dy^2} \: =0 Homework Equations Considering that \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}} how does one prove this statement...

Is it possible to find a directional derivative for a point on z = f(x,y) at a point (x,y) in a direction (u1,u2) using the plane tangent to z at (x,y)? If so, how? Thanks!

Homework Statement Prove that (for every x smaller than -1) \displaystyle \frac{1}{2}arctan\frac{2x}{1-x^{2}}+arccotx=\pi Homework Equations The Attempt at a Solution So i split the formula into two parts: \displaystyle \frac{1}{2}arctan\frac{2x}{1-x^{2}} and \displaystyle...

Homework Statement For \mu\geq 0, s\geq 1, prove that (1+s)^{\mu}\geq 1 + s^{\mu} Homework Equations The Attempt at a Solution I have written a proof involving the mean value theorem and derivatives, but there must be a simpler way! I think this should be done purely...

Hi, I remember having read in basic calculus that the following is true, but I don't know what this property is called and am having a hard time finding a reference to this. d u(x,y) = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy Ques: Is this true ? Is this true for...

Here is an example... d/dz(dp/dx) = d/dx(dp/dz) I don't remember learning this in my calculus or differential equations classes unless it was called something else.

I don't post many questions here since I'm usually able to find most of my answers before starting a thread. As only a few of you may know I'm gradually teaching myself calculus. What can I say? I guess I'm that kind of guy. I've managed to pick up on quite a few good basic ideas. There is...

Hi, I've begun learning about General Relativity, though I've already had some exposure to differential geometry. In particular, I understand that Lie Differentiation is a more "primitive" process than Covariant Differentiation (in that the latter requires some sort of connection). My...

I know that for any C2 function, the mixed second-order partials are equal, and I see that this should extend inductively to a statement about the kth partials of a Ck function, but I am having trouble figuring out exactly how this works. For example, take f:ℝ2 → ℝ . fxxy=fxyy is not true...

Homework Statement We want to construct a box with a square base and we only have 10 m2 of material to use in construction of the box. Assuming that all the material is used in the construction process determine the maximum volume that the box can have. Homework Equations Chain rule...

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 The Hyperbolic coordinate system is given by: u=2xy and v=x^2 - y^2 a.) Find the unit vectors u and v in terms of u,v,x(hat),y(hat) b.)Find the time derivatives of u(hat) and v(hat), your answers will have du/dt and dv/dt in them Homework Equations None really...

Homework Statement Let's say an object is moving with x(t)= e^(-1/t^2) it's motion is continuous everywhere and differentiable because its exponential so v(t)= 2e^(-1/t^2)/t^3 a(t)= e^(-1/t^2) *(4-6t^2)/t^6 I have asked to a calculus teacher and he said all the derivatives will be 0 at...

Homework Statement http://i.imgur.com/DQMRG.jpg Homework Equations The intervals are going by ones. The Attempt at a Solution Well for the first derivative, I'm guessing from -infinity to -1, it's a decreasing line? Also from -1 to 1...it's a constant negative line? I dunno