Higher-Order DifferentiationIn Exercises 23–26, find (a)
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Multivariable Calculus
- VECTOR DIFFERENTIATION:If A = t(^2)i - tj + (2t+1)k and B = (2t-3) i + j - tk, obtain the following at t=1:a) d/dt(A dot B)b) d/dt(A x B)c) d/dt |A + B| d) d/dt (A x dB/dt)arrow_forwardVECTOR DIFFERENTIATION: If R = e^(−t) i + ln(t^2+ 1) j - tant k. Find: (a) dR/dt, (b) d^2R/dt^2,(c) |dR/dt| ; (d) |d^2R/dt^2| at t = 0arrow_forwardF is the velocity field of a fluid flowing through a region in space. Find the flow along the given curve in the direction of increasing t. F =-y i + x j + 2 k r(t) = (-2 cos t)i + (2 sin t)j + 2t k, 0<= t<= 2paiarrow_forward
- Assuming that the equations in Exercises 35 and 36 define y as a differentiable function of x, find the value of dy/ dx at point P. 35. 1 - x - y2 - sin xy = 0, P(0, 1) 36. 2xy + ex+y - 2 = 0, P(0, ln 2)arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (7, 0), and (0, 4)arrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)arrow_forward
- Determine the total differential dH of the function:H(x,y,z) = x^y+zarrow_forwardF is the velocity field of a fluid flowing through a region in space. Find the flow along the given curve in the direction of increasing t. F = (x - z)i + x k r(t) = (cos t)i + (sin t)k, 0<=t<= paiarrow_forwardUse implicit differentiation to find dy/dx in Exercises 1–12. 1. x2y + xy2 = 6 3. 2xy + y2 = x + y 5. x2(x - y)2 = x2 - y2 7. y2 =( x - 1)/( x + 1) 9. x = sec y 11. x + tan (xy) = 0 2. x3 + y3 = 18xy 4. x3 - xy + y3 = 1 6. (3xy + 7)2 = 6y 8. x3 = (2x - y )/(x + 3y) 10. xy = cot (xy) 12. x4 + siny = x3y2arrow_forward
- (a) Let v(t) be a differentiable vector valued function of t. If v. (dv/dt) = 0 for all t, can we say anything about |v|?Justify your answer and give it a meaningful interpretation.arrow_forwardfind the linearization L(x, y) of the function ateach point. ƒ(x, y) = ex cos y at a. (0, 0), b. (0, π/2)arrow_forwardDescribe what it means for a vector-valued function r(t) to be continuous at a point.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning