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Explain why the function is differentiable at the given point. ROX. y) 6 + x In...
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not differentiable at (0,0) 4. You are told that there is a function F : R2 → R with partial derivatives F(x,y) = x2 +4y and Fy(x, y 3x - y. Should you believe it? Explain why. (Hint: use Clairaut's theorem)
Question 2...
showing multivariable calculus functions are differentiable
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2. Recall that by Theorem 3 of Section 14.3, a function f(x,y) is differentiable if its partial derivatives fa and fy both exist and are continuous. (a) Use this idea to show that the function f(x,y)-esin ry is differentiable. (b) Let o be a differentiable function and f(,)Jy Find the partial derivatives of f and determine whether they are continuous. Hint: The Fundamental Theorem of Calculus gives us that Ø has an...
2. Suppose the linear approximation of a differentiable function f(x, y, z) at the point (1,2,3) is given by L(x, y, z) = 17+ 6(x – 1) – 4(y – 2) + 5(2 – 3). Suppose furthermore that x, y and z are functions of (s, t), with (x(0,0), y(0,0), z(0,0)) = (1, 2, 3), and the differentials computed at (s, t) = (0,0) are given by dx = 7ds + 10dt, dy = 4ds – 3dt, dz = 2ds...
14. (-12 points] DETAILS SCALCETS 14.5.016. Suppose f is a differentiable function of x and y, and oC, s) - 9-s, 2 - 21). Use the table of values below to calculate g (5,2) and 9(5,2). f g 2 3 (43, -6) (5,2) fxy 1 8 7 9 3 2 9.(5,2) 9.(5,2) =
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
TICKET-IN-THE-DOOR Conceptual: Given a function z = f(x,y), write what the following notations mean in words. • 1x = 2t (partial denotion off wit x • ly =of sportial derivative off wit y) dy • 1. - 2f (2nd partial derivativt off wit > • ly = df (2nd portial derivative off wrt yo Practice: 1. Find the first-order partial derivatives of the following functions a. w -2xy + 5yVx+ 7y5 b. f(x,y) = ek + y Inx 2. Find...
6. A) Is the function y = |x + 5 continuous at x = -5? wh B) is the function y = |x + 5| differentiable at x = -5? why? 7. Use the rules of differentiation to find the derivative of the following functions. A) y = ln(2x + +5x + 7) B) y = tanx - COS X C) y = - - 355 D) y -
Suppose that f is a twice differentiable function and that its second partial derivatives are continuous. Let h(t) = f(x(t), y(t)) where x = 2e and y = 2t. Suppose that f:(2,0) = 4, fy(2,0) = 3, fx=(2,0) = 2, fyy(2,0) = 3, and fxy(2,0) = 2. Find out that when t=0.
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each of the variables is defined implicitly as a function of the others. 2 a) If F and z(x, y) are both assumed to be differentiable, fnd in terms of partial derivatives of F. b) Under similar assumptions on the other variables, find
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each...
y? - 2xy x + y2 if (x, y) + (0,0) 7. Given the piecewise function: f(x,y) 0 if (x, y) = (0,0) a) Show that: limf(x,y) does not exist. *(x,y) (0,0) b) Find: fy(0,0). c) Where is f continuous? Where is f differentiable? Explain.