Let F = (2y + 2)i +4ryj + rk. (a) Show whether F is conservative. (b) Using the fundamental theorem of calculus for line integrals, prove that F- dr = -a along a curve C defined by a vector function r(t) = cos ti + sin tj + tk for 0
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Q: 3. Let F = (2y² + z)i + 4.ryj + rk. (a) Show whether F is conservative. (b) Using the fundamental…
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Please answer (a),(b) and c in the following image.
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- Show that there are no analytic functions f = u + iv with u(x, y) = x² + y².Evaluate F - dr where F(x, y, z) = (4.25yz cos(ryz), 4.25xz cos(xyz), 4.25xy cos(ryz)) 1 to the point ( 7, 1 and C is the line segment joining the point (6, 2' 6 7' 6Let z be a differentiable function of r and y related by the following equation: x + ² =(√y + cosh(z)) ². x-Z Əz Find at the point (3,0,0). ду
- 4. Let f (x) = x³ -x² + 5. a) Find the y-intercept of f. y-intercept: b) Find f' and f", and determine where each are 0 and/or do not exist (DNE). If none, write "none". f' = 0: f' DNE: f" = 0: f" DNE: c) E Do a sign analysis on f' and f". d) Find the intervals on which f is increasing and decreasing. Increasing: Decreasing: e) Find the intervals on which f is concave up and concave down. Concave up: Concave down: f) answers as (x, y) points. Find all local maxima, local minima, and inflection points of f. Be sure to write your Local max: Local min: Inflection point(s): -4 -3 -1 g) Sketch the graph of f.Let u(t) = 21'i+ (-1)j-8k. Compute the derivative of the following function. (11 +7t) u(t) Select the corect 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+ i+ k. Click to select and enter your answer(s) and then click Check AnswerEvaluate along the curve y=x2 from (-1,1) to (2,4). First find the vector valued function r(t) defining the curve.
- Consider the function f(x) = x3 +4x² + 3. Find the points where the tangent line is horizontal. a) O (0,0) b) О (0, 3) 8 337 3' 27 3 1851 8' 512 c) O a o (G) (G) 33 131 2' 8 3 1851 8' 512 3 1797 8' 512 e) OEvaluate the integral of the vector functionConstruct an analytic function of the form f (z) = u + iv, where v is tan- (y/x), x # 0, y # 0