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Please show me all the steps to do this question. Thankyou! Lets : R → R...
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...
please show all work, even trivial steps. Here are definitions if needed. do not write in script thank you! 4. Letf: R2 → R2, by f(x,y) = (x-ey,xy). a. Find Df (2,0). b. Find DF-1(f (2,0)) Inverse Function Theorem: Suppose that f:R" → R" is continuously differentiable in an open set containing a and det(Df(a)) = 0, then there is an open set, V, containing a and an open set, W, containing f(a) such that f:V W has a continuous...
please help me answer All of these questions. Thank you! 1. Use the limit definition of the derivative - lift+A) 6 to find the derivative of the following function: f(x) = 3x-2x? You must SHOW YOUR WORK: 2. Find the slope of the tangent line at x = 4 (using the derivative you just found) 3. Using the point-slope form of the line, find the equation of the tangent line at x=4. Isolate the y. 1. Find the derivatives using...
Please help out is somewhat difficult. In practice, it is often easier to show a stronger condition: if each partial derivative OJi = 1, ... , n, is continuous in a disc around p = _ (a1.... , an), then f is differentiable дх, (a1,., an) Put differently: if f is continuously differentiable at p, it is differentiable at However, just as in the one-variable case, there are functions that are differentiable but not at p = p. continuously differentiable....
please solve the following question and show all the steps. Thank you. Question 2. We say that a function f is invertible if f--{(ba) : (a, b) function, in which case we call it the inverse function to f. Notice that f} is also a f- (b) = a-> b = f (a) (assuming that f-1 is a function). We define the range of a function f DR to be the set {f(x): r E D], i.e., the set of...
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).
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R defined by f(r,y)-+ (a) Show by explicit computation that the directional derivative exists at (x, y)- (0,0) for all oi rections u є R2 with 1 11-1, but that its value %(0.0) (Vf(0,0).u), fr at least one sucli u. (b) Show that the partial derivatives of f are not continuous at (0,0) if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R...
Consider the function Let where f(t) is differentiable for all t ∈ R. Show that z satisfies the partial differential equation (x2 − y2 ) ∂z/∂x + xy ∂z/∂y = xyz for all (x, y) ∈ R2 \ { (t, 0)|t ∈ R }.
showing multivariable calculus functions are differentiable Please help! 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...
Let A E(R") be Hermitian and positive definite, let v Define g R" R by R", and let cE R (a) Show that g is polynomial function of (... ,En) and in particular it has continuous partial derivatives of all orders. (b) Show that oo. Hint: Use Ezercise Ic. (c) Prove that g(x) achieves a global minimum d) Compute Vg(x). Show that g has a unique critical point, and hence argue that the minimum must be achieved at this point....