Determine an approximation for y = sin(x)
for 0 < X < 1 as
y ≈ a + bx + cx^2 + dx^3
dz Consider the equation 6 sin(x + y) + 2 sin (x +z)+ sin(y +z)= 0. Find the values of and dz ду at the point (41,41,- 3x). dx dz cx (Simplify your answer. Type an exact answer, using radicals as needed.) (41,4x - 3x) dz dy (43,4%, - 3x) (Simplify your answer. Type an exact answer, using radicals as needed.)
(a) (b) Find the least squares approximation of f(x) = x2 + 3 over the interval [0, 1] by a function of the form y = ae? + bx, where a, b E R. You should write the coefficients a, b as decimal approximations, rounded to two decimal places. Let g(x) be the least squares approximation you found in the pre- vious problem. So g(x) = ae” + bx for some scalars a, b. Find the least squares approximation of...
2 + COS- 2.ry dy d 1+y2 = y(y + sin x), 7(0) = 1. 3. [2cy cos(x+y) - sin x) dx + x2 cos (+²y) dy = 0. 4. Determine the values of the constants r and s such that (x,y) = x'y is an Integrating Factor for the following DE. (2y + 4x^y)dr + (4.6y +32)dy = 0. 2. C = -1 You need to find the solution in implicit form. 3. y = arcsin (C-cos) 4. r=...
explain please
2. Which one of the following DE is exact? a. (x+y)dx+(xy+1) dy=0 b (e + y)<x+ſe+x)dy = 0 c.(ye* +1) dx +(e' + xy) dy = 0 d. (sin x+cos y) dx +(cos x +sin y) dy = 0 e. (eº+1) dx +(e? + 2) dy = 0 3. The solution of the following separable DE xy' =-y? is a. y= '+c b. y=- c. y = In x+c In x+c d. In y=x? + e. yer+C 4....
Let y = sin(4x). If Ax = 0.3 at x = 0, use linear approximation to estimate Ay Ay = = Preview Find the percentage error rounded to 1 decimal place. error = Preview
(1 point) Find the Fourier approximation to f(x) = x over the interval (-11, ] using the orthogonal set {1, sin , cos x, sin 22, cos 2x, sin 3%, cos 3x}. You may use the following integrals (where k > 1): | 1 dx = 27 - x dx = 0 sin(kx) dx = 1 L z sin(kx) dx = (-1)k+1 cos(kx) dx =1 L", cos(kx) dx = 0 Answer: f(2) + 2/pi sin + -2/pi + + 0...
4. Determine the integral which computes the arc length of the curve y = sin(x) with 0 < x <. TT A '1 + sin2(a)dx so $." .TT B 1 + cos2(x)dx С [* V1 – cos? (7)dx D| None of the above.
Use the linear approximation of f(x) = sin(5x) at x = 0 to approximate sin(3). 15 3 6 20 4
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy. dz= Show that dz is an exact differential and determine the corresponding function f(x,y) Hence solve the differential equation = z sin( Cos( y) 2 x cos( y) dy 10] (b) Find the solution of the differential equation d2y dy 2 y e dx dæ2 initial conditions th that satisfi 1 (0) [15] and y(0) 0
4. (a Let (sin( x cos( ) dr...
[8] 2. Consider the differential equation dx + (1 - sin(v)) dy = 0 Determine if the equation is exact. If so, solve. If not determine an approximation integrating acco the equation exact. Verify that the new equation is exact, and solve the differential equation using the integrating factor you have found. (Hint: the integrating factor should be a function of y only.)