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- Find V. (V x F), if F(x, y, z) = 5e**i+ 5xe"j – 7ek. V·(V × F)Show that the line integral is independent of path. Jo 2xe-Ydx + (2y = x²e-⁄)dy, C is any path from (1, 0) to (5, 1) Ə The functions 2xe Y and 2y - x²ey have continuous first-order derivatives on R² and integral is independent of path. ay (2x -(2xe-x) = ду = Evaluate the integral. 25 e Ә Əx -(2y - x²e¯), so F(x, y) = )i + (2y − x²e¯½)j is a conservative vector field by the theorem given below, hence the line ӘР Əy Theorem: Let F = Pi + Qj be a vector field on an open simply-connected region D. Suppose that P and Q have continuous first-order partial derivatives and = ƏQ Əx throughout D. Then F is conservative.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.
- //welro) Find a local linear approximation for the function 5х + 6 f(x) valid for x near 5 5x + 6 Answer: Hint: The local linear approximation at x = 5 is the tangent line at x = 5. The equation for this tangent line, in slope-intercept form, is у 3 т(х — 5) +b where m f' (5) and b = f(5). So compute m and b, put these into the equation for the tangent line, and then enter the right-hand side of that equation as your answer.Find the interval on which the curve 1 y = dt 5+t+4t2 1 is concave upward. Note: When using interval notation in WeBWorK, you use I for o, -I for -0, and U for the union symba!. Interval =Let 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