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20) Draw a contour map of the function showing several level curves. f(x y) = x^3 − y
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- Let f(x,y)= x2-y2 a) On the set of xy-axes, draw the level curves L(x,y) = k for k= 1,2,3. On another set of axes, draw the level curves f(x,y)= k for k=1,2,3. (Do this part by hand) b) How do the contour maps in part (a) reflect the fact that L is the linear approximation to f at the point (2,1)? Explain briefly in words.How will the contour maps of f (x, y) = x and g(x, y) = 2x with contour interval 1 look different?The discriminant ƒxx ƒyy - ƒxy 2 is zero at the origin for each of the following functions, so the Second Derivative Test fails there. Determine whether the function has a maximum, a minimum, or neither at the origin by imagining what the surface z = ƒ(x, y) looks like. Describe your reasoning in each case. a. ƒ(x, y) = x2y2 b. ƒ(x, y) = xy2 c. ƒ(x, y) = x3y3 d. ƒ(x, y) = 1 - x2y2 e. ƒ(x, y) = x3y2 f. ƒ(x, y) =x4y4
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- Find the maximum and minimum values of the function f(x, y) = xy on the closed and bounded region defined as {(x, y) | x^2 + y^2 ≤ 1}. [Hint: You need to analyze the function first on the open region {(x, y) | x^2 + y^2 < 1} and then on the boundary {(x, y) | x^2 + y^2 = 1}.]Use Green's theorem to evaluate fxydx+ x3y3dy where C is a triangle with vertices (0,0), C (1, 0) and (1,2) with positive orientation. (b) Compute the directional derivative of the function f(x, y, z) = xy2 + y2z3 + z3x at the point P(4, -2, -1) in the direction of Q(l, 3, 2)Determine two functions, defined on the interval ( − ∞ , ∞ ) , whose Wronskian is given by W ( f 1 , f 2 ) = e 2 x . Are the functions that you found linearly independent on ( − ∞ , ∞ ) ? How do you know?
- Suppose that g is a function of two independent variables that has continuous partial derivatives, and consider the points P(8,6), Q(7,6), R(8,18) and S(6,7). The directional derivative of g at P in the direction of the vector PQ→ is 4, whilst the directional derivative of g at P in the direction of PR→ is 4. Find the directional derivative of g at P in the direction of the vector PS→. Give your answer correct to 2 decimal places. DPS→g(P)=prove that x^2y/(x^2+y^2) is not differentiable at 0,0 when x^2y/(x^2+y^2) if (x,y) not equal to (0,0) 0 if f(x,y) = (0,0)Determine two functions, defined on the interval (−∞,∞)(−∞,∞), whose Wronskian is given by W(f1,f2)=e2xW(f1,f2)=e2x. Are the functions that you found linearly independent on (−∞,∞)(−∞,∞)? How do you know?