- Compute Sy z(z−1)²(2²−1)dz where y is the circle |z – 2| = 5/2 with negative orientation.
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- Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.Evaluate ∫C (3y + exsinx) dx + (8x - ln(y3+2)) dy, where C is the circle x2 + y2 = 9 with positive direction using Green's Theorem.Let C be the circle given by x² + y² = 1. Evaluate √(-2x²y + cos x)dx + (2 xy² + y²e³)dy going in the counter-clockwise direction.
- O COSZ (ii) Sc (z 54 dz, where C is the circle z - = 1 %3D +-z) counterclockwise.(b) Sketch the graph of the function y = 1+ /16 – x²2 for -4 < x < 4. Hint: x2 + y? = 16 is the equation of a circle in the ry plane with center (0,0) and radius 4.1. Calculate y³ f. (1 simx) dx + (1-2) dy, + + y 3 3 where C is the circle of radius 1 centered at (0,0) oriented counterclockwise.
- c. Evaluate f 3x ds, where C consists of arc C₁ of parabola y = x² from (0,0) to (2,4), followed by the vertical line segment C₂ from (2, 4) to (2, 5). Sketch the path C and evaluate.-(2) Calculate ge yde + xzªdy - zy³ dz, where c is the circle x² +9² = 4, Z=3 (conterclockwise) hit: F = 2 cost i + 2 sint sint j + 3R | § J. dF = (OXT). R. A4. Let C be the circle |z| = 2 traversed once in the positive sense (counter clockwise). Compute : cos 3z dz 9c (z-1) (z-3)
- Let F = (7z+7x4) 7 + (3y + 2z + 2 sin (14)) 7 + (7x + 2y + 3e²¹ ) k. (a) Find curl F. curl F = (b) What does your answer to part (a) tell you about F. dr where C is the circle (x - 20)² + (y - 15)² = 1 in the xy-plane, oriented clockwise? JcF. dr = (c) If C is any closed curve, what can you say about F. dr? F. dr = (d) Now let C be the half circle (x - 20)² + (y − 15)² = 1 in the xy-plane with y > 15, traversed from (21, 15) to (19, 15). Find F. dr by using your result from (c) and considering C plus the line segment connecting the endpoints of C. JcF. dr =Find F. dr for given F and C. (a) (2,4). F = -y sin(x)i + cos(x)] and C is the parabola y = x² between (0,0) and (b) F = 2yi – (sin y)] counterclockwise around the unit circle C starting at the point (1,0).Graph the line that is described parametrically by (x, y) (2t,5 t), then.. %3D (a) Confirm that the point corresponding to t = O is exactly 5 units from (3,9). (b) Write a formula in terms of t for the distance from (3,9)to (2t, 5- t). (c) Find the other point on the line that is 5 units from (3,9). (d) Find the point on the line that minimizes the distance to (3,9). You may want a calculator.
