
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).
Real Analysis question, give clear writing please
Let h(x) be the function on (0, 1) defined by ſi x <1 h(x) = 2 X=1 (a) For any P, what is the value of L(f,P)? (b) Can you find a P such that U(f,P) is within 1/10 of L(f,P)? (c) Show that h is integrable.
9. La ste) defined as follows 9. Let f(x) defined as follows: f(x) = 0 if x < -1 = 2(x + 1)/27 if - 1<x<2 = 2/9 if 2 < x < 5 = 0 otherwise. Find F(u) = f(x)dx, where u E R.
show by
steps, definitions and theorems
" f(x) dx = 0 for all integers Let f(x) be a continuous function on (a,0). If n> 0, then show that f(x) = 0 on [a, b].
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
Let f : [0,∞) → R be the function defined by
f ( x ) = 2 ⌊ x ⌋ − x?
where x? = x − ⌊x⌋ is the decimal part of x. Prove that f is
injective.
Let f: 0,00) + R be the function defined by f(3) = 212) where ã = x — [x] is the decimal part of x. Prove that f is injective.
Q3 Given a function which is only defined for 0<x<5 and was found by modeling. Use it to compute the probability P(0<x< 1) after first normalizing to find the valid density 1+X NOTE: set up and do all integrations by hand, show all work and NOTE: SOL P[0<x<1) 33% Q4 Let X be the chi-squared density with k-6 degrees of freedom. Find the second moment of X, SOL E(X)=
Evaluate the piecewise-defined function. if x < 0 f(x) = { 3-X if os x<3 if x2 3 3 x + 3 (a) () (b) f(3) =
The f function differentiable at (-1,4) and 7(3) = 5 also let Hx f'(x) > -1. Find the greatest value f(0).
2. Consider the cubic spline for a function f on [0, 2] defined by S(x) = { ={ (z. 2x3 + ax2 + rx +1 if 0 < x <1 (x - 1)3 + c(x - 1)2 + d(x - 1) + ß if 1 < x < 2 where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.