If you differentiate (cot4 x)/4 you will get (cot3 x)(-csc x). The extra -csc x appears because of the chain rule. Note, however, it is missing in the integrant.   The correct way is to rewrite using the quotient identity and then the Pythagorean identity. Then we wind up being able...

You are better off with approximating it as it is a 4713-digit number:   In[7]:= 2212010 Out[7]= 16922717596544530505702408162179049678089581873189139797812419\ 2464100029210349272650642395169842735683318020409719814662260210702644\ 31715532311028263502380745286456...

We need ∇U = λ∇(px+qy) so the system is:   1/(2√𝑥) =  pλ 1/(2√y) = qλ px+qy=m   So we have:   q/√𝑥 = p/√y ⇒ q2y = p2x   (p+p2/q)x=m ⇒ p(q+p)x = mq ⇒ x=mq/[p(p+q)]   y = mp/[q(p+q)]   The...

sin(xyz)=x2y2 + z2 - 1   cos(xyz) (yz + xy ∂z/∂x) = 2xy2 + 2z ∂z/∂x   [xy cos(xyz) - 2z] ∂z/∂x = 2xy2 - yz cos xyz   ∂z/∂x = [2xy2 - yz cos(xyz)]/[xy cos(xyz) - 2z]   By symmetry:   ∂z/∂y = [2x2y - xz cos(xyz)]/[xy...

f(x,y) = xy/exp[(x2+y2)/2]   Find the gradient:   ∂f/∂x = {y exp[(x2+y2)/2] - x2y exp[(x2+y2)/2]}/exp(x2+y2)          = y(1-x2)/exp[(x2+y2)/2]   ∂f/∂y = x(1-y2)/exp[(x2+y2)/2]   So the gradient...

9x2 + 9y2 - 6x - 12y - 58 = 0   (9x2 - 6x + 1) + (9y2 - 12y + 4) = 63   (3x - 1)2 + (3y - 2)2 = 63   (x - 1/3)2 + (y - 2/3)2 = 7   So the center is (1/3,2/3) and the radius is √7.   The way I completed...

For the Euclidean N-gon, the sum is 180°(N-2) in degrees, or π(N-2) in radians.   So in your case, you get 180°·9 = 1620°.

We know that cosine is an even function, so therefore:   cos 6θ = cos 4θ   Case 1: 6θ = 2kπ + 4θ   2θ = 2kπ   θ = kπ   Case 2: 6θ = 2kπ - 4θ   10θ = 2kπ θ = kπ/5   Since the solution set...

x8 - 28x5 + 27x2 = 0   x2(x6 - 28x3 + 27) = 0   x2(x3 - 1)(x3 - 27) = 0   x2(x - 1)(x2 + x + 1)(x - 3)(x2 + 3x + 9) = 0   So setting each factor to 0 and solving gives all roots, real and complex.   Real: x...

We can use the adjugate matrix relation: A adj(A) = In det(A)   Then we have:   kA adj(kA) = In det(kA)   kA adj(kA) = kn In det(A)   adj(kA) = kn-1 A-1 det(A)   det(adj(kA)) = det(kn-1 A-1 det(A))   det(adj(kA)) = kn(n-1)...

Just let RHS be 0.   2y2 + 3y + 1 = 0   (2y + 1)(y + 1) = 0   y = -1/2, y = -1   We can also see these from the general solution to the ODE.   dy/dx = 2y2 + 3y + 1   dy/(2y2 + 3y + 1) =...

All you have to do is determine what the equation will be in each quadrant when the absolute values are removed.   To do this, note that |A| = A for all A ≥ 0, and |A| = -A for all A < 0.   You should get the following equations for the quadrants.   Q1:...

A) x = sin y + cos x   1 = (cos y) dy/dx - sin x   (cos y) dy/dx = 1 + sin x   dy/dx = (1 + sin x) / cos y     B) xy - y3 = sin x   y + x dy/dx - 3y2 dy/dx = cos x   (x - 3y2) dy/dx = -y +...

Problem 1. Note that the exact root is: x = - 3√4   Machine precision: -1.5874010519681994   Using seed x0 = -2:   f(x) = x3 + 4 so f'(x) = 3x2.   Thus we iterate: x ← x - (x3 + 4) / (3x2) = 2x/3 - 4/(3x2)   x0 = -2 x1...

Triple both sides and then regroup based on pairs which look like a product rule.    3(x2 - 4xy - 2y2)dx + 3(y2 - 4xy - 2x2) dy = 0   3y2 dy - (6y2 dx + 12xy dy) - (12xy dx + 6x2 dy) + 3x2 dx   x3 - 6x2y - 6xy2 + y3 = C &n...

f(x) = 4 - 3x - x2   [f(x) - f(a)] / (x - a)   = [(4 - 3x - x2) - (4 - 3a - a2)] / (x - a)   = (-3x - x2 + 3a + a2) / (x - a)   = [-3(x - a) - (x + a)(x - a)] / (x - a)   = -3 - x - a

(z2 - 3z + 2)(z2 + 6z + 8) / (z2 + 3z - 4)(z2 - 4)   = (z-1)(z-2)(z+2)(z+4) / (z+3)(z-1)(z+2)(z-2)   = (z+4) / (z+3)

You can only give bounds for this number.   Let's assume, the n hexagons must form a polyhex (like a polyomino but with regular hexagons).   Maximum:   The first hexagon contributes 6 vertices and each new hexagon after that contributes at most 4 vertices.   MAX...