Investigation Consider the graph of the vector-valued function
(a) Approximate the length of the curve by finding the length of the line segment connecting its endpoints.
(b) Approximate the length of the curve by summing the lengths of the line segments connecting the terminal points of the
(c) Describe how you could obtain a more accurate approximation by continuing the processes in parts (a) and (b).
(d) Use the
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Calculus: Early Transcendental Functions
- Identities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3. ∇ x (F x G ) = (G ⋅ ∇) F - G (∇ ⋅ F) - (F ⋅ ∇)G + F (∇ ⋅ G)arrow_forwardIdentities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3. ∇ x (∇ x F) = ∇(∇ ⋅ F) - (∇ ⋅ ∇)Farrow_forwardIdentities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3. ∇ ⋅ (F x G ) = G ⋅ (∇ x F) - F ⋅ (∇ x G)arrow_forward
- Identities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3. ∇ (F ⋅ G ) = (G ⋅ ∇) F + (F ⋅ ∇)G + G x (∇ x F) + F x (∇ x G)arrow_forwardNonuniform straight-line motion Consider the motion of an object given by the position function r(t) = ƒ(t)⟨a, b, c⟩ + ⟨x0, y0, z0⟩, for t ≥ 0,where a, b, c, x0, y0, and z0 are constants, and ƒ is a differentiable scalar function, for t ≥ 0.a. Explain why r describes motion along a line.b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?arrow_forwardInterpreting directional derivatives Consider the functionƒ(x, y) = 3x2 - 2y2.a. Compute ∇ƒ(x, y) and ∇ƒ(2, 3).b. Let u = ⟨cos θ, sin θ⟩ be a unit vector. At (2, 3), for what values of θ (measured relative to the positive x-axis), with 0 ≤ θ < 2π, does the directional derivative have its maximum and minimum values? What are those values?arrow_forward
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