(a) the stationary paths of the functional | S[y] = [d dx F(x,y,y', y″), y(a) = A, y(b) = B, satisfy the Euler-Lagrange equation ම‍ෙ d² OF dx2 ay" ;) d 'ƏF` dx dy' მყ OF + = 0, y(a)=A,_y(b) = B, Fy"|q = Fy" |₁ = 0. (b) Apply the result found in part (a) to the functional defined in equation (10.1) (page 219), with p = constant and the boundary conditions y(0) = y(L) = 0, to derive the associated Euler-Lagrange equation and show that its solution is y(x) = pg -x (L − x) (L² + xL — x²) . 24k - -

1) Find the linearization L(x) of f(x) at x = a.f(x) = 2x2 + 4x - 1, a = 4 2) Use implicit differentiation to find dy/dx and d2y/dx2.xy + 3 = y, at the point (4, -1) 3) Provide an appropriate response.The line that is normal to the curve x2 - xy + y2 = 25 at (5, 5) intersects the curve at what other point? 4) Find the second derivative.y = 4x2 + 10x + 2x-3 5) For the given function, find each of the following:(A) Intercepts(B) Vertex(C) Maximum or minimum(D) Rangef(x) = (x + 1)2 - 4 6) Find the slope of the line tangent to the curve y = 3x2 + 9x - 5 at the point (-2, -11). 7) Use implicit differentiation to find dy/dx.cos xy + x4 = y4 Pleas i want answer this Questions (1 -7) Analytical Geometry & Calculus

show that the L operator is linear for y" – 4y" + 2y' +y = x?

6. Let f(x,y) = In(x² + y² +1) + e²x". (a) Find the gradient of f at the point (0,–2) (b) Find the directional derivative of f at the point (0,–2) in the direction of (5,–12) (c) Find the maximum value of the directional derivative at the point (0,-2)

Let a ≤ b, A and B be constants. S[y] = [° da (y^² +w²y² + 2yxª¹), y(a)=A, _y(b)= B, a where w is a positive constant. the stationary path of S[y] satisfies the Euler-Lagrange equation, d'y dx² -w²y = x², y(a) = A, y(b) = B, For the case a = 0, b = 1, find a stationary path of S[y] for the boundary conditions y(0) = −24/w6, y(1) = 0.

Find the linearization of the function f(x,y) = y + sin (x/y) at the point (pi,3)

B) Find the derivative of the foiiowing: 1) y = 2sin2(5x) ,2) y = x, 3) xsin-14y = y tan-14x %3D 4) y = ecosh(8x) ,5) y = t2 and x = V4t + 1 t2 + 1 x°cosx 6) y = Vtan?x, 7) y = ln |V1 + x2

Let F(t) = (e' cos t) î+ (e' sin t) ĵ + e' k. (a) Find symmetric equations of the line tangent to F at the point P(1,0, 1). (b) If f(t) = e2t, compute (fF) (0).

Show that f fdx + gdy = S - (16x°y?dx + 8yx*dy) is independent of path: • (10,5) (1,–2) • (10,5) -2) fe ду dg •(10,5) -2) – (16a*y dx + 8yæ*dy) = 3

Find the gradient of cos(xy) – 3y4 + 3 at the point (1,)