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yIP is paralle FaPat P df be the point at which f has a local extreme value. Th...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
The parametric curve r=(2t2+8t−5,−2cos(πt),t3−28t)r=(2t2+8t−5,−2cos(πt),t3−28t) crosses itself at one and only one point. The point is (x,y,z)=(x,y,z)= ( , , ). Let θθ be the acute angle between the two tangent lines to the curve at the crossing point. Then cos(θ)=cos(θ)= (1 point) The parametric curve r (2128t 5,-2 cos(t), 281) crosses itself at one and only one point. The point is (x.y,z- Let 0 be the acute angle between the two tangent lines to the curve at the crossing point....
Question 2, non-calculator Question 1, calculator The curve C in the x-y-plane is given parametrically by (x(t), y(t), where dr = t sine) and dv = cos| t The Maclaruin series for a function f is given by r" for 1 sts 6 a) Use the ratio test to find the interval of convergence of the Maclaurin series for f a) Find the slope of the line tangent to the curve C at the point where t 3. b) Let...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0 Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2-2.2-b with b є R. (a) Prove that the tangent line of each curve in H at a point (x, y) with y 0 has slope - (b) Let y-f(x) be a...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2b with b ER. (a) Prove that the tangent line of each curve in H at a point (r, y) with y / 0 has slope (b) Let y -f(x) be a...
full workings required Let f: R^2 → be a differentiable function and let CCR be a curve in R^2 described by the cartesian equation f(x,y) = Letla.b) R be a point that lies on the curve Cck and assume that the partial derivatives off evaluated at (a,b) satisfy: fr(a,b) 0 and fy(a,b) +0. Also assume that there exists an expression y-g(x) that solves the equation f(xxx)=0 fory in terms of x in a neighbourhood of the point (8.b). This means...
-n ', S Let f(x,yZFz2_xy. Let v=<1,1,1>. Let point P=<2,1,3> a. Compute gradient of fx,y,z) b. If the contours are far apart, is the length of the gradient large or small? Answer: Explain! What MATLAB command is used to draw the gradient vectors? Answer: - c. Compute the directional derivative in the direction of v. d. Compute the equation of the tangent plane to f(x,y,z) at the point P. e. Use the chain rule to compute r if x t2,...
true or false is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If a unit vector u makes the angle of π/4 with the gradient ▽f(P), the directional derivative Duf(P) is equal to |Vf(P)I/2. F 12. There is a point on the hyperboloid 2 -y is parallel to the plane...
Exercise 7.4.* Suppose that at a Jacobian and gradient given by point , a nonlinear equality constrained problem has 1 -3 -3 6 8 -1 -8 and Vf() ()f -6 2 0 2 -7 2 Does t satisfy the first-order necessary conditions for a local constrained minimizer? If not, find unique vectors g^ E null(J(r)) and gn E range(J( such that Vf(x) Hence find a vector direction p such that Vf (x)"p < 0 and J(f)p = 0. = gR...