Higher-Order DifferentiationIn Exercises 23–26, find (a)
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Multivariable Calculus
- find 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_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_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 k * only d ,e, f *arrow_forward
- Using 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_forwardFind the linearization of the function f(x,y) = y + sin (x/y) at the point (pi,3)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
- Explain why the function is differentiable at the given point. Then find the linearization L(x,y) of the function at that point. f(x,y)=y+sin(x/y),(0,3)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_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_forward
- Determine the total differential dH of the function:H(x,y,z) = x^y+zarrow_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_forwardapply linearity property of laplace in the equation f(t)= t cos 2t and f(t)=t^2e^-3tarrow_forward
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