* if <<1 Let h(x) = { 2 – 22 if i< x < 2 2 - 3 if < > 2 Use the limit definition of derivative to find h'(1) if it exists.
Problem 10. Let f,g: [a,b] -R be Riemann integrable functions such that f(x) < g(x) for all x E [a,b]. Prove that g(x)
+1 4. Consider the function ISO 0<<1 -1 = 1 0 1<*52 (x - 2)2 => 2 (a) (10) Use the definition of the limit of a function at a point to evaluate with proof (b) (10) Use the definition of continuity at a point to prove that /(x) is not continuous at -1. (e) (2) Is /(x) uniformly continuous on (-1,2)? If it is, prove it. Other- wise, explain why not. (d) (8) Is f() uniformly continuous on (1,3)?...
(c) [5 points] Prove that f(r) [5 p ) = Σ (-1-rn oints Prove that f(x converges uniformly on [-c, c when 0<c<1. lenny
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
1. For pdf f (r, y) = 1.22, 0 < x < 1,0 < y < 2, z +y > 1, calculate: EY) and () E (X2)
A periodic function f(x) with period 21 is defined by: X + -1<x< 0 2 f(x) = 0<x< 2 Determine the Fourier expansion of the periodic function f(x). X - TT
Now assume that f(0) = 0 and f'(0) = 0. Prove that if f is twice differentiable and If"(x) < 1 for all x E R then 22 Vx > 0, f(x) < 2
1. Let x, a € R. Prove that if a <a, then -a < x <a.
Let X and Y have join density 6 f(x, y) =-(x + y)2, 0 < x < 1, 0 < y < 1