Find the value of a so that the work done by the force ♬ = (y² + 1)₁ + (x + y)ĝ in moving from (0,0) to (1,0) along the curve y = ax(1 - x) is a minimum. 5 19 At α = 2 where W min or 15 8 15 are both wrong! O Fractions please:)
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- Find the work done by F (given below) when moving the particle CCW around T once. T is the triangle with points (2,2), (0,0), and (2,0). F(x,y) = sinx i + x^2y^3 j. Show this done using green's theorem and w/o Green's theorem1. Let R and b be positive constants. The vector function r(t) = (R cost, R sint, bt) traces out a helix that goes up and down the z-axis. a) Find the arclength function s(t) that gives the length of the helix from t = 0 to any other t. b) Reparametrize the helix so that it has a derivative whose magnitude is always equal to 1. c) Set R = b = 1. Compute T, Ñ, and B for the helix at the point (√2/2,√2/2, π/4).A particle is under the influence of the force F= (-cosh (4x4) + xy)i + (e-y + x)j. The corner points move once in the counterclockwise direction on the rectangular curve in (1.1), (1.7), (3.1) and (3.7). Find the work done.
- 5. Let C be the portion of the parabola y = x², oriented from the starting point (0, 0) towards the end point (3,9). Find the unit tangent vector T and the unit normal vector n to C at the point (1, 1).Please type only the answers. Do not handwritten. Thank you!Consider the curve C with parametrisation where t E R. r(t) = = cos(5t) cosh(t) i+ sin(5t) cosh(t) -j+tanh(t)k, (a) Show that C lies on the surface of the sphere described by the equation x² + y² + z² = 1. (b) Find the vector equation of the line L that is tangent to C at the point r(0).
- Which of the vector functions below represents the curve of intersection of the following two surfaces: x² +4y²+4z² = 16 and y = x². Select one: ○ a. r (t) = (t, t², O b. r (t) = (t, t², |○ c. r (t) = (t, t₁, ○ d. r (t) = t, -t² = (t, 16++²-4+4 4 1 16-12-4t 4 O e. None of the above ○ f. r (t) = (t, t², √ 16-12+4t 4 16-12-4t 4 4-12-4t4 4Q8..