Homework Statement
Hello all, thank you for the help in advance. It's a two-sided derivative problem, for lack of a better term, and I appreciate all hints or help. If we have a function y so that
y=bx for all x<0, and
y= x^2-13x for all x> or = 0,
for what value of b is y differentiable at...
Is the time derivative of a curl commutative? I think I may have answered this question.... Only the partial time derivative of a curl is commutative? The total time derivative is not, since for example in cartesian coordinates, x,y,and z can themselves be functions of time. In spherical and...
Homework Statement
So the first part asks to prove the time derivative of kinetic energy is dT/dt=F dot product v which I did not problem. but then the second part of the problem asks to prove that if the mass is changing with time then the time derivative of d(mT)/dt=F dot product m and i'm...
When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##?
\frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...
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
For f(x) = abs(x^3 - 9x), does f'(0) exist?
The Attempt at a Solution
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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
Let A be a set of critical points of the function f(x).
Let B be a set of roots of the equation f''(x)=0.
Let C be a set of points where f''(x) does not exist.
It follows that B∪C=D is a set of potential inflection points of f(x).
Q 1: Can there exist any inflection points...
Homework Statement
1) I am having trouble with the questions, "Use the logarithmic derivative to find y' when y=((e^-x)cos^2x)/((x^2)+x+1)
Homework Equations
(dy/dx)(e^x) = e^x
(dy/dx)ln(e^-x) = -x ?
The Attempt at a Solution
First I believe I put ln on each set of terms (Though I don't know...
Homework Statement
The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp?
Homework Equations
r=pq
The Attempt at a Solution
I substituted...
Homework Statement
Show that if F is an antiderivative of f on [a,b] and c is in (a,b), then f cannot have a jump or removable discontinuity at c. Hint: assume that it does and show that either F'(c) does not exist or F'(c) does not equal f(c).
2. The attempt at a solution
I attempted a proof...
Hi,
So I'm working through a bunch of problems involving gradient vectors and derivatives to try to better understand it all, and one specific thing is giving me trouble.
I have a general function that defines a change in Temperature with respect to position (x,y). So for example, dT/dt would...
Homework Statement
A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°?
This is the diagram...
Homework Statement
Suppose a can, after an initial kick, moves up along a smooth hill of ice. Make a statement concerning its acceleration.
A) It will travel at constant velocity with zero acceleration.
B) It will have a constant acceleration up the hill, but a different constant acceleration...