Surface
46. F = 〈e–y, 2z, xy〉 across the curved sides of the surface
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Calculus: Early Transcendentals (3rd Edition)
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- 38. Motion along a circle Show that the vector-valued function r(t) = (2i + 2j + k) %3D + cos t V2 j) + sin t V2 j + V3 V3 V3 describes the motion of a particle moving in the circle of radius 1 centered at the point (2, 2, 1) and lying in the plane x + y – 2z = 2.arrow_forwardStokes' Theorem (1.50) Given F = x²yi – yj. Find (a) V x F (b) Ss F- da over a rectangle bounded by the lines x = 0, x = b, y = 0, and y = c. (c) fc ▼ x F. dr around the rectangle of part (b).arrow_forwardYouTrylt 1 Find an equation of the tangent plane to the given surface at the specified point. Surface: z =/xy, Point: (1,1,1)arrow_forward
- Find an equation for the plane that is tangent to the given surface at the given point. z= √y-x (0,1,1) Find the equation for the tangent plane to the surface z = √y-x at the point (0,1,1). Use a coefficient of 1 for x. = 0arrow_forwarda) A three dimensional motion of an object is given by the vector function r(t) = 4 cos t i+ 4 sin tj+5 k. Sketch the motion of the object when 0arrow_forwardFind the (linear) equation of the tangent plane to the surface z=xsin(x+y) at the point (7,−7,0). Your answer should be in the form of an equation, e.g., something like ax+by+cz=d or a(x−x0)+b(y−y0)+c(z−z0)=0 would work. Equation: ____________arrow_forwardWhich of the following best describes the surface with the vector equation r(u, v) = (cos u)i + (sin u)j + vk? %3D O A plane passing through the origin O A cylinder whose axis is the z-axis O A sphere of radius 1 centered at the origin O A right circular cone whose vertex is the originarrow_forwardFind an equation of the tangent plane to the surface at the given point. 4x? + 2y? + 4z? = 22, P = (2, 1, 1) (Express numbers in exact form. Use symbolic notation and fractions where needed. Let f(x, y, z) and give the equation in terms of x, y, and z.) equation:arrow_forwardLet (P) be a plane considered as a surface in the space, parameterized by X(u, v) = (u, v, au + bv + c) where a, b, and c are all constants, with c + 0. Then: The tangent plane at each point is perpendicular to (P) The normal vector varies constantly The above answer The above a nswer The second fundamental form equals e The second fundamental form is zero The above answer The above a ns werarrow_forwardX Complete the following steps to find an equation of the tangent plane to the surface z = the point (1, 2, 2): Step 1: Determine the function F(x, y, z) that describes the surface in the form F(x, y, z) = 0: Step 2: Find F(x, y, z) Step 3: Find Fy(x, y, z) Step 4: Find F₂(x, y, z) = Step 5: Find F(1, 2, 2): = Step 6: Find Fy(1, 2, 2) = Step 7: Find F₂(1, 2, 2): 1 X -1 -2 1 = -1 2 F(x, y, z) y X 11 Hon The equation of the tangent plane at the given point is 2x -y-=-2arrow_forwardFind the point(s) on the surface at which the tangent plane is horizontal. z = 8 - x² - y² + 7y Step 1 The equation of the surface can be converted to the general form by defining F(x, y, z) as F(x, y, z) = 8 - x² - y² + 7y - z The gradient of F is the vector given by VF(x, y, z) = F Fx(x, y, z)= = II Step 2 Determine the partial derivatives Fx(x, y, z), F(x, y, z), and F₂ 11 əx = -2x X (8 - x² - y² + 7y - z) (x, y, z)i + F₂(x, y, z)=(8 - x² - y² + 7y − z) -2y -1 -2x Ə F₂(x, y, z) = (8 (8 - x² - y² + 7y - z) əz -2y -1 |(x, y, z)j + F₂(x, y, z)k. y +7 F₂(x, y, z).arrow_forwardKalan süre 1:11:41 Let z = g(x, y) = f(3 cos(xy), y + e™Y) provided that f(3, 4) = 5, f1(3, 4) = 2, f2(3, 4) = 5. i) Find g1 (0, 3). ii) Find g2 (0, 3). iii) Find the equation of the tangent plane to the surface z = point (0, 3). f(3 cos(xy), y + e=Y) at the Türkçe: f(3, 4) = 5, f1(3, 4) = 2, f2(3, 4) = 5 olmak üzere z = g(x, y) = f(3 cos(xy), y + e"Y) olsun. %3D 6. i) 91 (0, 3) değerini bulunuz. ii) g2 (0, 3) değerini bulunuz. iii) z = f(3 cos(xy), y + e#Y) yüzeyine (0, 3) noktasında teğet düzlemin denklemini bulunuz. O i) 15, ii) 5, iii) 15x + 5y - z = 10 о) 15, i) 5, iї) 15х - 5у - z %3D0 о ) 15, i) 15, iil) 15х + 15у + z%3D40 O ) 45, ii) 5, iii) 45x - 5y - z = -35 O i) -30, ii) 15, iii) -30x + 15y + z = -20 О i) 45, i) -10, iil) 45х -10y -z%3D-20 O i) -15, ii) -10, iii) -15x -10y - z = 10 O i) -30, ii) 20, iii) -30x + 20y - z = 10arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage