Consider the vector field F and the curve C below. F(x, y) = (3 + 8xy²)i + 8x²yj, C is the arc of the hyperbola y = 1/x from (1, 1) to (4,1) (a) Find a potential function f such that F = Vf. f(x, y) = (b) Use part (a) to evaluate Jo Vf. dr along the given curve C.
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- Let u(t) = 9t°i+ (P - 2)j-3k and v(t) = e'i+6 ej- ek. Compute the derivative of the following function. u(t) • v(t) Select the correct choice below and fill in the answer box(es) to complete your choice. O A. The derivative is the scalar function O B. The derivative is the vector-valued function Di+ ( Dj+ k.5cost i+ 3sintj, sint 9. Given the vector-valued functions, (a) a(t) (b) a(t) = 5 tan(t) i + cosec(t)j + 10t?k ,Find a(0), a) and a() Do any of these functions have domain restrictions?Sketch the curve represented by the vector-valued function r(t) = (t + 1)i + (3t − 1)j + 2tk and give the orientation of the curve.
- Use the equation giving the flux of the vector field across the curve to calculate the flux of F(x, y) = (e", 4x – 1) across C, the parabola y = x for 0 < x < 9, oriented left to right. (Use symbolic notation and fractions where needed.) F. dr = Ttterf (9i) - 2025 IncorrectA charged particle begins at rest at the origin. Suddenly, a force causes the particleto accelerate according to the vector function a(t) = ⟨ sin(t) , 6t , 2cos(t)⟩Find functions for the velocity, speed and position of the particle at time tLet u(t) = 5t°i+ 2-9)j-8k and v(t) = e'i+9 e-j- e k. Compute the derivative of the following function. u(t) • v(t) Select the correct choice below and fill in the answer box(es) to complete your choice. The derivative is the vector-valued function (i+ ( i+ (k. O B. The derivative is the scalar function