b) Show that F - dr is independent of its path, given that y F(r, y, 2) = 1+. + tan-2) i+ tanrj+ k. Hence, find the potential function of F and then evaluate the work done by F moving an object along the smooth curve C, the line segment from (1,0, –1) to (0,0, 0) then along the parabola z = y,r = 0 to (0, –1, 1).

Q: Let  f(x, y) = sin(15x2 + 3y2) Calculate the quadratic approximation of f at the point (0,0)

A: Given that,  fx, y=sin15x2+3y2 The quadratic approximation of the function fx, y at the point x0, y0…

Q: 4. (a) Determine the directional derivative of f(x, y, z) = zety at the point (1, −1, 1) in the…

A:

Q: An equation for the tangent plane to the surface : = (xty at the point P(1, 1, 4) is a) 8x + 8y -: =…

A: Equation of a tangent to the surface z=f(x,y) at point (a,b,f(a,b)) is given by…

Q: Evaluate | f(x,y) ds where Cis the portion of the curve y = In x from the point (1,0) to the point…

A:

Q: One of the following vector functions is a gradient Select one: O (xy? + 2y?)7+ (2y³ – x²y+ 2x)j O…

A:

Q: 2. Determine the directional derivative of o(x, y, z) = re" + y² + xyz at the point (2, 0, 3) in the…

A:

Q: 8. Find the directional derivative off at the p in the direction of v = () of f(x, y, z)=x²exx, p =…

A:

Q: 5. nts) Let F =, use Green's Theorem to evaluate F. dr where the curve C is C the curve shown in the…

A:

Q: Find the implicit equation of the tangent plane to z = e + x + x² + 12 at the point (6, 0, 55).

A: Given: z=ey+x+x2+12 Point (6,0,55)

Q: TRUE OR FALSE. A plane curve C defined by the parametric equations x = f(t) and y = g(t), where t is…

A: check the given statement is true or not.

Q: 6. Given f(x, y) = 2x³y=x²y³, a) find the directional derivative of f(x, y) at (1, 1) in the…

A:

Q: Consider the function f(r, y) = e=² sin( q² + y²) (a) ) Compute the gradient of f. (b) 1 ) Find the…

A: Given fx,y=e-xsinx2+y2

Q: a) Describe the curve obtained when we make y = 2 and 2= √2 a f b) Represent on this curve the…

A: Given That : y=2 and z=2 To Find : a) Describe the Curve obtained when y=2 and z=2 b) Represent on…

Q: determine wave direction in the function f(x,y) x^y — x²y³ decreases faster at the point (2,-3)

A: Wave direction in the function f(x,y)= x4y-x2y3 that decreases faster at the point (2,-3) is…

Q: 10. Given the curve C defined by the vector-valued function r(t) = a) find an equation of the line…

A:

Q: The function f(x, y) = ¿e=y – In(y²)

A: It is partial derivative

Q: 18.) Compute the directional derivative of f in the given ✓ given points P. 3 a) f(x, y, z) = xy² +…

A:

Q: Compute the directional derivative of f(r. y) = = at the point (1, 1) in a irection normmal to the…

A:

Q: Find the equation of the tangent line to right branch of the hyperbola x = sec t, y tant, -sts at…

A: Given query is to find the equation of tangent line.

Q: Suppose F(x, y) = x² + y²j and C is the line segment segment from point P = (-3,-4) to Q = (-3, 1).…

A:

Q: a) Find the directional derivative of the function f(x, y, z) = x2 y (2z+1)² at the point P:(1, -1,…

A: This question is related to gradient and it's properties.....

Q: Let C be the curve connecting (0,0,

A: Given that C be the curve connecting  (0, 0, 0) to (1, 4, 1) to (3, 6, 2) to (2, 2, 1) to (0, 0, 0).…

Q: a) Compute the gradient of the function f(x, y, z) = e*y² . b)Compute the directional derivative Daf…

A:

Q: The graph of y=x'-3x-24x +32 is concave upward on the interval (1, oo) because: O'=y"=0 Oy' is…

A: The given function is y=x3-3x2-24x+32

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.