We have seen that all
[Hint: Let G(x, y, z) = ⟨g(x, y, z), 0, 0⟩, where
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Multivariable Calculus
- Suppose that the directional derivative of a function f at a point Pin the direction of the vector ū = (-1,0) is - Relate this 4 value to the behaviour of f. A. From the point P, the function f is increasing at a rate of - in the positive x-direction. B. From the point P, the function f is increasing at a rate of - in the positive y-direction. C. From the point P, the function f is decreasing at a rate of - in the negative y-direction.arrow_forwardIf is a smooth curve given by a vector function r(t) a < t < b and is a constant vector, show that fc v dr - v [r(b)- r(a)]arrow_forwardThe nabla formulas corresponding to the derivative rule of income are gVf+fVg, (Vƒ). F + f(V.F), (Vf) x F + f(V x F). ▼(fg) V. (fF) ▼ x (fF) = = = Justify the middle one (e.g. in the case of n = 3). Hint: The coordinate functions of the vector field fF are fF₁ etcarrow_forward
- Find the limits of the following vector functions. ((-3) 2t³-t² tan (2t)\ 21² (i) (ii) lim In 1-0 5t lim (-+ 3.-306 (6) (+3,- ²+6arrow_forwardPlease solve part (c) of the problem.arrow_forwardCompute the derivative of the vector valued functions. (a) Calculate the derivative in two different ways to verify the rules in Section 3.2: r (g (t)) where r (t) = (2 sin 2t, 6 cos 2t) and g (t) = t (b) Calculate the derivative in two different ways to verify the rules in Section 3.2: ri (t) · r2 (t) where r; (t) = (t°, t°, 4t) and r2 (t) = (t-1,1+ t, 2)arrow_forward
- Correct solution will be liked.arrow_forwardCompute ∇ƒ(-1, 2, 1), where ƒ(x, y, z) = xy/z.arrow_forwardA net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v=(x-y,z+y+7,z2) and the net is decribed by the equation y=1-x2-z2, y20, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)arrow_forward
- Suppose that f(z,y) is differentiable, and suppose that the directional derivative of f at the origin attains a maximum value of 5 in the direction of the vector from the origin to the point (-3,4). Find Vƒ(0,0).arrow_forwardGiven the vector-valued function below, solve for the following:a. Domain of Rb. Show that R is continuous at t = 2.c. If f(t) = t^2, evaluate (R◦f)'(-1)arrow_forward10. Given the curve C defined by the vector-valued function F(t) = a) find an equation of the line that is tangent to this curve at the point (1,1,0). (You can give the parametric equations of the line.) b) Find the length of the curve C from the point (1,1,0) to the point. (1,-1,5) (in the direction of increasing t). c) Compute the curvature of the curve C at the point (1,1,0). 11. Determine whether the lines L₁ given by x=t, y=1+2t, z=2+ 3t, (with -∞ < t < oo) and L₂ given by x = 3 - 4s, y = 3+2s, z = 2-s, (with -∞ < s <∞o) are parallel, intersect or are skew lines. 12. Consider the function f(x, y) = { 27² if (x, y) = (0,0) 1 if (x, y) = (0,0) a) Does lim(z,y)+(0,0) f(x, y) exist? Justify your answer. b) Is f continuous at (0,0)? Justify your answer. af c) Compute of (0,0), (0,0), if they exist there. (Hint: You will need to use the limit definition of the partial derivative.)arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage