3. Let f(r) be defined by and let F(x) be defined by F(x) = Í f()...
Let y'(x)y(x)g'(x) = g(x)g'(x), y(0) = 0, x e í, where f'(x) denotes ar(X) and g(x) is a given non- 4. dx constant differentiable function on R with g(0) = g(2) = 0. Then find the value of y(2) Let y'(x)y(x)g'(x) = g(x)g'(x), y(0) = 0, x e í, where f'(x) denotes ar(X) and g(x) is a given non- 4. dx constant differentiable function on R with g(0) = g(2) = 0. Then find the value of y(2)
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
Problem 2 (5 points) Let f be a continuous function over R, and let g(x) represent a differentiable function such that 8(2)=- Given that the relationship dt = 29(x)-7 is true for all x, find the following. a) Value of g(1); (2 pts) b) Value of (2). (3 pts)
true or false The real valued function f : (1,7) + R defined by f(x) = 2is uniformly contin- uous on (0,7). Let an = 1 -1/n for all n € N. Then for all e > 0) and any N E N we have that Jan - am) < e for all n, m > N. Let f :(a,b) → R be a differentiable function, if f'() = 0 for some point Xo € (a, b) then X, is...
1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove: f is differentiable at -2 f is differentiable at 1 2) Prove the product rule. Hint: Use f(x)g(x)− f(c)g(c) = f(x)g(x)−g(c))+f(x)− f(c))g(c). 3) Prove the quotient rule. Hint: You can do this directly, but it may be easier to find the derivative of 1/x and then use the chain rule and the product rule. 4) For n∈Z, prove that xn is differentiable and find the derivative, unless, of course, n...
5. Let be the function defined by f(x) = -1 3 1.5 if r <0 if 0<x<2 if 3 < r <5 Find the Lebesgue integral of f over (-10,10).
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) =...
Let f : Rn × Rn → R be the inner product function: f(r,y)-(2,3) 1. Using the definition of multivariable derivative, calculate D fab and the Jacobian matrix f'(a, b) 2. If f, g : Rn → R are differentiable and h : R → R is defined by h(t)-(f(t), g(t)), show that 3. If f : R → Rn is differentiable and Ilf(t)ll = 1 for all t, show that(f,(t)T,f(t))-0
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentiable everywhere and calculate its total derivative (b) If f is differentiable everywhere and Df (x)A, for some A E Mp and all q.p x E Rp, show that there exists b E R, such that f(x) = Ax + b for all x E Rp 3. Let f: RP-R (a) If f(x)-Ax + b,...
(2) Consider the function f : R → R defined by Í 1 x E [-L,0) f(x + 2L) = f(z) -(x) f( 2L) o E l0,L) a. Graph f on the interval [-3L, 3L]. b. Compute Fi-L,Lf) c. Graph F-L(f) on the interval [-3L,3L] c. Graph Fi-L/2,L/() on the interval [-3L,3L]. (2) Consider the function f : R → R defined by Í 1 x E [-L,0) f(x + 2L) = f(z) -(x) f( 2L) o E l0,L) a....