Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists...
Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists for every u 70, but f is not differentiable at (1,1). Prove that your choice of , has these properties
Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists for every u # 0, but f is not differentiable at (1,1). Prove that your choice of f has these properties
2. Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists for every u # 0, but f is not differentiable at (1,1). Prove that your choice of f has these properties (8 points).
Analysis on Metric Spaces Find a function f : R2 + R such that the partial derivative f'((1,1); u) exists for every u 70, but f is not differentiable at (1,1). Prove that your choice of f has these properties (8 points).
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
3. (a) Let f be an infinitely differentiable function on R and define F(x) = [-vf(u) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cea for sufficiently large n. Find the least n for which it is true.
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...
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that 2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'(x)|< A, for all x E R". Prove that the function F(x)= x - f(x) maps R" one-to-one and onto R". (Suggestion: Use the Contraction Mapping Principle Why not use the Inverse Function Theorem?) differentiable function and there exists 0