4. (a) Evaluate feeds where C is the are of the parabola y = r² from (0,0) to (1,1) (b) Evaluate fyz cos x ds where C is given by x = t, y = 3 cost, z = 3 sint, 0 tπ. (c) Evaluate foxy dx + y² dy+yz dz where C is the line segment from (1,0, -1) to (3, 4, 2). 5. Determine if the following vector fields are conservative or not. If it is conservative, find the potential function f such that Vf = F

Let Q = 4x2 + 6xy+ 4y? . The points on the curve Q = 7 which are closest to the origin are: %3D %3D OA(-눈-)and () (주·주) pue OB. (-V7, V7) and (v7, -V7) Oc. (0, ±1) OD. (tV7,0)

Draw a diagram to show that there are two tangent lines to the parabola of y = x2 that pass through the point (0, -16). Find the coordinates of the points where these tangent lines intersect the parabola. smaller x-value (x, y) = larger x-value (x, y) =

5.Find the point on the parabola x? = 4y nearest to (0,4). (x.y)

Q1: A) Find the nearest distance from the point (-4,2) and the line 2x – y = 5 d²y dy and dx? B) Find if x = sin 3z and y = tan23z dx Q2: A) Find an equation of the line tangent to the function of x2 + (y - x)² = 9 at x = 1. dy if y = dx x-x3 B) Using the definition of derivative, find 1-x

The plane x = 1 intersects the paraboloid z = x2 + y2 in a parabola. Find the slope of the tangent line to the parabola at (1, 2, 5)

Suppose a ball is hit with a horizontal velocity of 80 ft/sec and an initial upward velocity of 90 ft/sec. The parametric equations (without regard for air resistance) for position (x,y) at time t seconds are: x(t)=80t y(t)=90t−16t^2 d. A 25-foot high fence is at x=400 ft. According to this parametric function, will the ball go over the fence, hit the fence, or hit the ground before reaching the fence? e. At what time does the ball reach its maximum height ? What is the horizontal distance at this time?

Evaluate √(2² + yz sin(xyz))dx+(y²+xz sin(xyz))dy+(x+xysin(xyz))dz where C is the curve following the outline for the triangle from (1,0,0) to (0,1,0) to (0, 0, 1) and back to (1,0,0).

Let L be the line with parametric equations x = 2+2t y = 3 z = 4+3t Find the shortest distance d from the point Po=(-3, -3, 4) to L, and the point Q on L that is closest to Po. Use the square root symbol 'V' where needed to give an exact value for your answer.