Evaluating a function In Exercises 11 and 12 evaluate the
(a)
(b)
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Chapter 12 Solutions
Calculus
- Finding a Least Approximation In Exercises 75-78, a find the least squares approximation g(x)=a0+a1xof the function f, and b use a graphing utility to graph fand gin the same viewing window. f(x)=sinxcosx, 0xarrow_forwardVerifying InequalitiesIn Exercises 53-64, verify a the Cauchy-Schwarz Inequality and b the triangle inequality for given vectors and inner products. Calculusf(x)=sinx, g(x)=cosx, f,g=0/4f(x)g(x)dxarrow_forwardFinding a Least Squares Approximation In Exercises 73-76, a find the least squares approximation g(x)=a0+a1x+a2x2of the function f, and b use a graphing utility to graph fand gin the same viewing window. f(x)=sinx, /2x/2arrow_forward
- Linear TransformationsIn Exercises 9-22, determine whether the function is a linear transformation. T:R2R2, T(x,y)=(x,y2)arrow_forwardFinding an Image and a PreimageIn Exercises 1-8, use the function to find a the image of v and b the preimage of w. T(v1,v2)=(32v112v2,v1v2,v2) v=(2,4), w=(3,2,0)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
- Using a Function (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f (x, y) = c at P, (c) find the tangent line to the level curve f (x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = 9x2 + 4y2, c = 40, P(2, −1)arrow_forwardAlgebra find the laplace transform of x(t) = sin(t) [u(t) - u(t-2π)] h(t) = [u(t+2π)- u(t)] determine: y(t) = x(t) * h(t) sketch the grapharrow_forwardSplitting a vector field Express the vector field F = ⟨xy, 0, 0⟩in the form V + W, where ∇ ⋅ V = 0 and ∇ x W = 0.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_forwardGradient fields in ℝ3 Find the vector field F = ∇φ for thefollowing potential functions. φ(x, y, z) = 1/ | r | , where r = ⟨x, y, z⟩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 (F x G ) = (G ⋅ ∇) F - G (∇ ⋅ F) - (F ⋅ ∇)G + F (∇ ⋅ G)arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage