(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....
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Assume that there exists a function L : (0, oo) → R such that L'(x) = 1/x for x > 0, Calculate the derivatives of the following functions: (a) f(x) := L(2x + 3) for x > 0, g(x) := (L(H)), for x > 0, k(x) := L(L(x)) when L(x) > 0, x > 0. (b) (c) h(x) := L(ax) for a > 0, x > 0, (d)
Exercise 3. Let f : [0,1]- R be non-negative and Riemann integrable. Assume of()dr 0. otherwise. Show that g is not Riemann integrable Exercise 3. Let f : [0,1]- R be non-negative and Riemann integrable. Assume of()dr 0. otherwise. Show that g is not Riemann integrable
Exercise 7.9. Assume f:R → R. (a) Let t € (1,0). Prove that if |f(x) = alt for all x, then f is differentiable at 0. (b) Let t € (0,1). Prove that if f(x) = |x|* for all x, and f(0) = 0, then f is not differentiable at 0. (c) Give a pair of examples showing that if |f(x)= |x|for all I, then either conclusion is possible.
3. Let f: R+R be a function. (a) Assume that f is Riemann integrable on [a, b] by some a < b in R. Does there always exist a differentiable function F:RR such that F' = f? Provide either a counterexample or a proof. (b) Assume that f is differentiable, f'(x) > 1 for every x ER, f(0) = 0. Show that f(x) > x for every x > 0. (c) Assume that f(x) = 2:13 + x. Show that...
Exercise 41.2 Consider the signal f(t) window w(t)e amat, α E R, and the Gaussian (a) Verify that is well-defined (even though f f L'(R)). (b) Compute Ws(A, b) using the following result: e-ra(1+iz)2 dt = a-2 For a > 0 and E R, (c) Show that l W, (A, b)12 attains its maximum when λ a. Exercise 41.2 Consider the signal f(t) window w(t)e amat, α E R, and the Gaussian (a) Verify that is well-defined (even though f...
Problem 4 if the Wronskian W off and g is 3e**, and if f(x) = e2*, find g(x).
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
I. Let f : R2 → R be defined by f(x)l cos (122) 211 Compute the second order Taylor polynomial of f near the point xo - 0. A Road Map to Glory (On your way to glory, please keep in mind that f is class C) a) Fill in the blanks: The second order Taylor's polynomial at h E R2 is given by T2 (h) = 2! b) Compute the numbers, vectors and matrices that went into the blanks...