Let C₁ be the directed line segment that starts at (2,0) and ends at (-3,-5) and C₂ ɔe the parabola with parametrization x = t, y = 4 – t² for t € [−3, 2]. Let ♬ be the vector field F(x, y) = (x² − 4y) î + (2x + 3y²) î. (a) Evaluate Sc₂ F · dr. (b) Evaluate §♂ Ē·dr where C' is the positively oriented path consisting of the parabola y = 4—x² starting from (2,0) to (−3,−5) and the line segment from (−3,−5) back to (2, 0). (c) Use the results in (a) and (b) to evaluate Sc₂ F · dr.

1.) is the quantity r(t+h)-r(t) a vector or a scalar? identify this object in the applet. 2.) is (r(t+h)-r(t))/h a vector or a scalar? Describe what represents r(t+h)-r(t)/h 3.) slide h toward to 0. How does r(t+h)-r(t) change? How about (r(t+h)-r(t))/h?

A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v=(x-y,z+y+7,z2) and the net is decribed by the equation y=1-x2-z2, y20, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)

Suppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect.

Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2.r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2πThen the vector-valued functiondescribes a tilted circle (a circle in a plane that is not parallel to one of the coordinateplanes). The center of the circle is O(0, 0, 0).(a) Show that |r(t)| is constant and determine the constant. This is the radius of thecircle.(b) Find an equation for the plane that contains the circle by doing the following:(i) Find three points on the circle.(ii) Use the three points to find a normal vector to the plane containing the three points.(iii) Find an equation for the plane. Check: does the plane contain the center of the circle?(iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)

How would I write a vector function r(t) that follows the below guidelines

3 - Find the Derivatives of the following Vectors, Where F(t) = 2secti – 3j* + csctk and G(t)* = 3ti – t²j - 4csctk a) F× G* b) F.G→