Using a FunctionIn Exercises 43–46, (a) find the gradient of the function at
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Chapter 13 Solutions
Multivariable Calculus
- Hooke’s Law In Exercises 65-68, use Hooke’s Law, which states that the distance a spring stretches (or compresses) from its natural, or equilibrium, length varies directly as the applied force on the spring. A force of 220 newtons stretches a spring 0.12 meter. What force stretches the spring 0.16 meter?arrow_forwardUsing Properties of the Derivative In Exercise 26, use the properties of the derivative to find the following. (a) r′(t) (b) d dt [u(t) − 2r(t)] (c) d dt [(3t)r(t)] (d) d dt [r(t) ∙ u(t)] (e) d dt [r(t) × u(t)] (f) d dt [u(2t)] 26. r(t) = sin ti + cos tj + tk, u(t) = sin ti + cos tj + 1 t karrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (6, 0), and (0, 6) a) 216 b) 72 c) 0 d) 36arrow_forward
- Find the gradient of the function f(x,y,z)=y^2ln(xz), at the point (1,2,e) ∇f(1,2,e)=arrow_forwardGradient fields on curves For the potential function φ and points A, B, C, and D on the level curve φ(x, y) = 0, complete the following steps.a. Find the gradient field F = ∇φ.b. Evaluate F at the points A, B, C, and D.c. Plot the level curve φ(x, y) = 0 and the vectors F at the points A, B, C, and D. φ(x, y) = y - 2x; A(-1, -2), B(0, 0), C(1, 2), and D(2, 4)arrow_forwardExercise. Find the equations of the tangent and the normal at the point indicated. 1. y = 3x2 -2x + 1 at (1, 2). 2. y = 2 + 4x - x2 at x= -1. 3. x2 + y2 - 6x + 2y = 0 at (0, 0). 4. y = x2 - 2x at its points of intersections with the line y = 3. 5. a2y = x3 at (a, a).arrow_forward
- Describe what it means for a vector-valued function r(t) to be continuous at a point.arrow_forwardSketch the plane curve represented by the vector valued function and give the orientation of the curve. r(t) = (t2 + t)i + (t2 − t)jarrow_forwardHeat flux in a plate A square plate R = {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has a temperature distribution T(x, y) = 100 - 50x - 25y.a. Sketch two level curves of the temperature in the plate.b. Find the gradient of the temperature ∇T(x, y).c. Assume the flow of heat is given by the vector field F = -∇T(x, y). Compute F.d. Find the outward heat flux across the boundary {(x, y): x = 1, 0 ≤ y ≤ 1}.e. Find the outward heat flux across the boundary {(x, y): 0 ≤ x ≤ 1, y = 1}.arrow_forward
- Eavluate the gradient vector at the given point. a) f(x,y) = cos2(πxy) at (1,-1) b) f(x,y,z) = xeyz at (2,1,0)arrow_forwardCurve In Exercise 56, sketch the plane curve and find its length over the given interval. 56. r(t) = t 2i + 2tk, [0, 3]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_forward
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