Could you rephrase your question with clarity, please?

2τ = 0.02 sec

DEFINITION ::
Let X and Y be sets of real numbers. A real-valued function of a real variable, x from X to Y is a correspondence that assigns to each number x in X exactly one number y in Y.

Why did you use the multiplication operator *?

Where did the negative sign come from?

Which differentiation formula did you apply that comes with the negative sign?

Typo error. Moreover, it doesn't show the full workings because the author assumes all readers have strong algebra basics. But the graph is correctly sketched. Perhaps you should show your workings here.

∂soo,

Your answer −x² is correct. My point is, you must have a basis to find the anti-derivative of −2x if you want to evaluate ∫ (−2x) dx. Please share with me, what Differentiation Rule did you use? Type out the formula in LaTeX for clarity. The more you figure out yourself, the better you learn math. I'm just guiding you.

I tend to use these two for spacing:

x \quad y →

x\; y →

http://en.wikipedia.org/wiki/Help:Displayi...formula#Spacing

Not 216 N? What is the answer?

You don't need to find the angle of inclination θ because you can determine cos θ and sin θ from the Pythagorean theorem. The hypotenuse is 20 m and altitude (height) is 1 m.

ninty and ystiang have already covered this.

Where did you get this question at A-level? This one is interesting because it is a Multivariable Optimization problem with Equality Constraint, which is considered an Advanced Math topic. In standard form, the problem is to



where (x, y) is a 2-dimensional vector called the design vector, f(x,y) is termed the objective function, and g(x,y) = 0 is known as the equality constraint. There are few methods to solve it such as the popular Lagrange multiplier method and the Direct Substitution method, but I prefer the Solution by the method of Constrained Variation. The basic idea is to find a closed-form expression for the first-order differential of f(df) at all points at which the constraint is satisfied. The desired optimum points are then obtained by setting the differential df equal to zero, that is introduced without proof:



Equation (2) can be rewritten as



So we have



Equation (3) gives



The optimum values of x can be obtained substituting Eq.(4) into Eq.(2):



The optimum values of y can be obtained from Eq.(4):



The optimum values of function f can be obtained from Eq.(1):



This solution gives the minimum value of x² + y² as



REMARK ::
-------------
Please note that f(x,y) = x² + y² is an infinite paraboloid in a 3-D plot. The constraint g(x,y) is the dark blue circle in the contour plot. The objective is to find the minimum value of f(x,y) around the constrained geometry g(x,y). From the contour plot, you can also estimate the maximum value of f(x,y) occurs at somewhere near x = -10 and y = 25 (see the lighter band).



This post has been edited by Critical_Fallacy: Jan 28 2014, 09:00 AM

Using SPM Calculus, you can solve it by Direct Substitution.

If we choose to eliminate y from Equation (1): f(x,y), then Equation (2): g(x,y) gives



Thus the objective function f(x,y) becomes



Plotting Equation (6a):



Plotting Equation (6b):



The desired optimum points are then obtained by setting the derivative df/dx of Eq.(6a) & Eq.(6b) equal to zero.



The optimum values of y can be obtained from Eq.(5):



The optimum values of function f can be obtained from Eq.(1):



This solution gives the minimum value of x² + y² as

The basic idea of the trigonometric solution is to convert Cartesian coordinate (x, y) to Polar coordinate (r, θ), from 2 variables to 1 variable. Pretty genius! You have learned this already in Critical Tutorial #3: Complex Numbers.







Solution 1: long version







Solution 2: short version

恭喜发财，马到成功！

To solve this problem, you have to develop the equations of motion which require a relationship between the forces acting on a particle and the accelerated motion they cause.



The general technique is to write down the equation you get by applying Newton’s 2nd law to the combined particle ‘car + caravan’, and then solve the equation for the unknown. To do that, you are advised to draw separate force diagrams for car and caravan. Drawing this diagram is very important since it provides a graphical representation that accounts for all the forces which act on the particle, and thereby makes it possible to resolve these forces into their x, y components in 2-D.



It is also common to assume in a model like this that the tow bar is inextensible. That means it will not stretch. For mathematical convenience, we generally assume that the sense of each acceleration component acts in the same direction of as its positive inertial coordinate axis (denoted by the right-arrow notation → for horizontal axis, and the up-arrow notation ↑ for vertical axis).



I do one example (Part B). We are mainly concerned with the horizontal components because we want to determine the tension T. Ask Krevaki or kingkingyyk to check!



This post has been edited by Critical_Fallacy: Feb 3 2014, 09:19 PM

CLASSIC PROBLEMS ::

The main objective is to help you to develop the ability to analyze classic problems in a simple and logical manner and to apply to its solution a few, well-understood, basic Physics principles.

Question 1

The two blocks shown start from rest. The horizontal plane and the pulley are frictionless, and the pulley is assumed to be of negligible mass. Determine the acceleration of each block and the tension in each cord.



(Ans :: a_A = 8.40 m/s², a_B = 4.20 m/s², T_ADC = 840 N, T_CB = 1680 N)

Question 2

The bob of a 2-m pendulum describes an arc of circle in a vertical plane. If the tension in the cord is 2.5 times the weight of the bob for the position shown, find the velocity and the acceleration of the bob in that position.



(Ans :: a_T = 4.90 m/s², a_N = 16.03 m/s², v = ±5.66 m/s)

Question 3

Determine the rated speed of a highway curve of radius r = 125 m banked through an angle u = 18°. The rated speed of a banked highway curve is the speed at which a car should travel if no lateral friction force is to be exerted on its wheels.



(Ans :: 71.8 km/h)

By the definition of asymptotes, when both x and y become very large, the equation approximates to (also can be rearranged to ), since .

But using the difference of squares formula, is also the equation of two straight lines and .

So, for large values of x and y, the graph approximates to a pair of straight lines. Hence, the pair of lines and are the asymptotes of the graph.

Hi delsoo,

If you look closely at the equation , y tends to become very large when x approaches to 0, because . Therefore, the third asymptote is a vertical line .

You are correct! The length of the hypotenuse is 20 m. I overlooked this when I sketched the diagram.

Obviously the graph has only one real root (or repeated root) at x = 0. The question does not specify the number of real roots. The solution given is a graphical method. If you sketch the graph nicely and understand how a function works, this approach is easy.





Otherwise, go for the analytical method in Post #870.

This post has been edited by Critical_Fallacy: Feb 6 2014, 10:17 AM

This is analytical method:


If a (coefficient of x²) is nonzero, the condition for having real roots:


Find the k-values that satisfy the inequality.


Results:


But what defines a (coefficient of x²) is k − 1. Since a must be nonzero, therefore

Interesting question from VengenZ! Must invite kingkingyyk, Krevaki, ninty, v1n0d, ystiang.

It can be shown that



and



I think the question is to find



This post has been edited by Critical_Fallacy: Feb 6 2014, 04:56 PM

Spoon-feed is to feed (someone) by using a spoon. But I feed using thought-provoking words.

Perhaps you can imagine the collisions between billiard balls. If they “collide elastically” the colliding particles do not stick together. What do you mean by “move together”?