a) The cycloid's curve is given by
and
Differentiate x with respect to phi to obtain
Differentiate y with respect y to obtain
Hence
This can be simplified to
Now use the identity
to obtain
Now use the identity
to obtain
Now take the square root on both sides
and integrate to obtain s
b) The potential energy with C = 0 becomes
Now use the identity
to obtain
Now plug in
and write U in terms of s
(a) Using the Pythagorean Theorem, we can state that Show that d)d) and (b) We need...
Consider the vector field F(x, ) (4x3y -6ry3,2rdy - 9x2y +5y*) along the curve C given by r(t)(tsin(rt), 2t +cos(xl)), -2ss 0 To show that F is conservative we need to check a) b) We wish to find a potential for F. Let r,y be that potential, then Use the first component of F to find an expression for ф(x, y)-Po(x,y) + g(y), where ф(x,y) in the form: Differentiate ф(x,y) with respect to y and determine g(y) e Using the...
we want to implement the function ?(?3,?2,?1,?0)=∑?(1,4,5,8,11,12,13). Assume complemented inputs are available at no cost. c. Use Shannon’s Expansion Theorem to implement H using a 2-to-1 multiplexer with z3 as the select line, AND gates, and OR gates. Also, show the expression for H that uses Shannon’s Expansion Theorem. d. Use Shannon’s Expansion Theorem to implement H using a 4-to-1 multiplexer with z2 and z1 as the select lines. Also, show the expression for H that uses Shannon’s Expansion Theorem.
2. (18 marks total) In this exercise, we will derive the famous "envelope theorem". Suppose you wish to (unconditionally) maximize some objective function f(x,y; a), where r and y are two variables you can choose, while a is some variable that is given exogenously. Note that, even though we don't get to choose a, it may still affect the optimal choice of r. An example of a variable like this would be the wage in the household problem we discussed...
e. Application of Bayes: Using Bayes theorem driven in item d, show that in the following problem switching is the better strategy for the player and makes the probability of wining as 2/3, while not switching gives the probability of wining 1/3. Monty Hall: There are 3 doors 1, 2, 3. Randomly, and equally likely, behind one of the door there is a prize, and other two are empty. The player choose one door. He does not open the door...
3. In this problem we shall investigate the intermediate value theorem for derivatives. (a) Differentiate the function f(c)= sin ), 2 0 = 0,1=0 Show that f'(0) exists but that f' is not continuous at 0. Roughly sketch f' to see that nevertheless, f' doesn't seem to "skip any val- ues". Now let f be any function differentiable on (a, b) and let 21,22 € (a, b). Suppose f'(21) < 0 and f'(22) > 0. (b) By the Extreme Value...
can you please prove the following theorem using the provided axioms and defintions. using terms like suppose in a paragraph format. please write clearly or type if you can ! 1 Order Properties Undefined Terms: The word "point and the expression "the point z precedes the point y will not be defined. This undefined expression wil be written z < y. Its negation, "z does not precede y," will be written y. There is a set of all points, called...
I know Graph 1 is not conservative and Graph 2 is conservative but how can we find vector function F for Graph 2? Because F is deliberately not given. Project 1. Fundamental theorem of line integrals amenta al theorem of line integrals: if F is a In our course we learned the conservative vector field with potential f and C is a curve connecting point A to b, then F dr f(B) f(A). Moreover it happens if and only if...
2. (a) We want to find the root x of the function f(x); that is, we need f(r) = 0 . This can be done using Newton's method, making use of the iterative formula f(xn) Show that the sequence ofiterates (%) converges quadratically if f'(x) 0 in some appropriate interval of x-values near the root χ 9 point b) We can get Newton's method to find the k-th root of some number a by making it solve the non-linear cquation...
using final theorem: : How did we get the above with the equation given below(please show all steps so I can get it right) : C(2) c(o) lim Z (R(z)R(z) 0.02341z0.0219 c(o)lim Z 72-1.7953z+0.8406/(z1 z-1 0.02341+0.0219 0.46z+0.33 C(2) 0.46z+0.33 z2-1.368z+ 0.368 0.46z+0.33 1+ - 1.368z+ 0.368 z2 0.908z +0.698 R(z) 0.46z 0.33- 1 0.908z-10.698z-
can anybody explain how to do #9 by using the theorem 2.7? i know the vectors in those matrices are linearly independent, span, and are bases, but i do not know how to show them with the theorem 2.7 a matrix ever, the the col- ons of B. e rela- In Exercises 6-9, use Theorem 2.7 to determine which of the following sets of vectors are linearly independent, which span, and which are bases. 6. In R2t], bi = 1+t...