Using a Function Consider the function
(a) Sketch the graph of f in the first octant and plot the point (1, 2, 4) on the surface.
(b) Find
(i)
(c) Find
(i)
(iii) v is the
(iv) v is the vector from
(d) Find
(e) Find the maximum value of the directional derivative at (1, 2).
(f) Find a unit vector u orthogonal to
Want to see the full answer?
Check out a sample textbook solutionChapter 13 Solutions
Calculus
- Walking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a. In each case, find z'(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = x2 + 4y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2πarrow_forwardChange of variables Consider the parameterized curvesr(t) = ⟨ƒ(t), g(t), h(t)⟩ and R(t) = ⟨ƒ(u(t)0, g(u(t)), h(u(t))⟩, where ƒ, g, h, and u are continuously differentiable functions and u has an inverse on [a, b].a. Show that the curve generated by r on the intervala ≤ t ≤ b is the same as the curve generated by R onu-1(a) ≤ t ≤ u-1(b) (or u-1(b) ≤ t ≤ u-1(a)).b. Show that the lengths of the two curves are equal.(Hint: Use the Chain Rule and a change of variables in the arc length integral for the curve generated by R.)arrow_forwardAverage temperature Consider a block of a conducting material occupyingthe region D = {(x, y, z): 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 1}.Due to heat sources on its boundaries, the temperature in the block is given by T(x, y, z) = 250xy sin πz. Find the average temperature of the block.arrow_forward
- Derivative rules Suppose u and v are differentiable functions at t = 0 with u(0) = ⟨0, 1, 1⟩, u'(0) = ⟨0, 7, 1⟩, v(0) = ⟨0, 1, 1⟩, and v'(0) = ⟨1, 1, 2⟩ . Evaluate the following expression.arrow_forwardTangent planes Find an equation of the plane tangent to the following surfaces at the given points. e xy2z3 - 1 = 1; (1, 1, 1) and (1, -1, 1)arrow_forwardTangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t. x = cos t + t sin t, y = sin t - t cos t; t = π/4arrow_forward
- Integral Hyperbolic functions and identities #11,12arrow_forwardTopic: Manifolds and Differential Forms ( Advance Multivariable Calculus) Q. Let α = x1 dx2 + x3 dx4, β = x1 x2 dx3 dx4 + x3 x4 dx1 dx2 and γ = x2 dx1 dx3 dx4 be forms on R4 . Calculate (i) αβ, αγ; (ii) dβ, dγ; (iii) ∗α, ∗γ.arrow_forwardVariable density The density of a thin circular plate of radius 2 is given by p(x, y) = 4 + xy. The edge of the plate is described by the parametric equations x = 2 cos t, y = 2 sin t, for 0 ≤ t ≤ 2π.a. Find the rate of change of the density with respect to t on the edge of the plate.b. At what point(s) on the edge of the plate is the density a maximum?arrow_forward
- Circulation in a plane A circle C in the plane x + y + z = 8 has a radius of 4 and center (2, 3, 3). Evaluate ∮C F ⋅ dr for F = ⟨0, -z, 2y⟩, where C has a counterclockwise orientation when viewed from above. Does the circulation depend on the radius of the circle? Does it depend on the location of the center of the circle?arrow_forwardVolume of a solid obtained by revolving about the x-axis the region below the graph of y=sin x cos x over the interval [0, pi/2].arrow_forwardCurve sketching show all work f(x) = sin(x)cos(x) on [-pi, pi]arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage