Divergence of a Cross Product In Exercises 73 and 74, find
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Chapter 15 Solutions
EBK CALCULUS: EARLY TRANSCENDENTAL FUNC
- Nonuniform 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_forwardFind the linearization L(x,y) of the function f(x,y)arrow_forwardTangent lines Suppose the vector-valued function r(t) = ⟨ƒ(t), g(t), h(t)⟩ is smooth on an interval containing the point t0. The line tangent to r(t) at t = t0 is the line parallel to the tangent vector r'(t0) that passes through (ƒ(t0), g(t0), h(t0)). For each of the following functions, find an equation of the line tangent to the curve at t = t0. Choose an orientation for the line that is the same as the direction of r'.arrow_forward
- Tangent lines Suppose the vector-valued function r(t) = ⟨ƒ(t), g(t), h(t)⟩ is smooth on an interval containing the point t0. The line tangent to r(t) at t = t0 is the line parallel to the tangent vector r'(t0) that passes through (ƒ(t0), g(t0), h(t0)). For each of the following functions, find an equation of the line tangent to the curve at t = t0. Choose an orientation for the line that is the same as the direction of r'. r(t) = ⟨3t - 1, 7t + 2, t2⟩; t0 = 1arrow_forwardFind r(t)u(t). r(t) = (3-7)i + +³1 + u(t) = t²i - 8j + t³k r(t)u(t)= j+ 4k Is the result a vector-valued function? Explain. Need Help? Yes, the dot product is a vector-valued function. O No, the dot product is a scalar-valued function. Watch Itarrow_forwardFind the domain of the vector function r(t) = } Domain: {tarrow_forwardFind the maximum rate of change of f(x, y, z) = x+y/z at the point (3, 4, 5) and the direction in which it occurs. Maximum rate of change: Direction (unit vector) in which it occurs: Carrow_forwardFind the maximum rate of change of f(x, y, z) = x + y/z at the point (1, 1, –5) and the direction in which it occurs. Maximum rate of change: Direction (unit vector) in which it occurs:arrow_forwardFind r(t) · u(t). r(t) = (5t – 3)i + t³j + 2k u(t) = t2i – 6j + t3k r(t) · u(t) = Is the result a vector-valued function? Explain. Yes, the dot product is a vector-valued function. No, the dot product is a scalar-valued function.arrow_forward(b) Consider the vector-valued function r(t) = t 2 i + (t − 3)j + tk. Write a vector-valued function s(t) that is the specified transformation of r. iii. a horizontal translation 5 units in the direction of the positive y-axisarrow_forwardVector-valued functions (VVF) VVF Point P t value (for point 1 r(t) = ½ t²i + √4 − t j + √t + 1k (0, 2, 1) Graph the tangent line together with the curve and the points. Copy your code and the resulting picture. Use MATLAB (Octave) code for graphing vector valued functions. A) 2arrow_forwardDisplacement d→1 is in the yz plane 62.8 o from the positive direction of the y axis, has a positive z component, and has a magnitude of 5.10 m. Displacement d→2 is in the xz plane 37.0 o from the positive direction of the x axis, has a positive z component, and has magnitude 0.900 m. What are (a) d→1⋅d→2 , (b) the x component of d→1×d→2 , (c) the y component of d→1×d→2 , (d) the z component of d→1×d→2 , and (e) the angle between d→1 and d→2 ?arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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