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Explain the key points and reasons for the establishment of 12.3.2(
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- Determine if the following statements are true or false. If it is false, explain why?28.P. Let f. (x)= |x]*1f for x in [-1, 1]. Show that each f. is differentiable on [-1, 1] and that (f.) converges uniformly on [-1, 1] to f(x)= x]. 11+1/nShow that (the picture) converges uniformly on R to a differentiable function f which satisfies |f(x)| ≤ |x|, |f'(x)| ≤ 1 for all x ∈ R.
- Explain the key points and reasons for the establishment of 11.3.2(integral Test)1. Show that the function f(x) = 1/³ is uniformly continuous on the set [1,00), but that is not uniformly continuous on (0, ∞).Construct a sequence of continuous functions fn : R → R such that 0 < fn s 1, limn0 Jo fndx = 0 but that the sequence fn() converges for no x € [0, 1].
- A college professor always finishes his lectures within 2 minutes after the bell rings to end the period and the end of the lecture. Let X = the time (in minutes) that elapses between the bell and the end of the lecture and suppose the pdf of X is: f(x) = (0 x² for 0 ≤ x ≤2 otherwise Compute the exact probability (using integral calculus) that the lecture continues for at least 60 seconds beyond the bell.Use extrapolation to derive an O(h) formula for f'(xo). O Suppose that N(h) is an approximation to M for every h > 0 and that M = N(h) + K,h + K2h + K,h+ , for some constants K1, K2, K3, Use the values N (h), N (), and N () to produce an O(h) .... approximation to M.2
- Consider the "floor" function f given by f(x) = [x], where [x] is the immediate integer preceding x. Is the integral r+∞ [ f(x) dx convergent?(Boredom killer) Define r(x) = / to-le-t dt. It can be shown that the function is well-defined for positive x, the right converges for positive x. Find I(n) for all natural numbers n. that is, the integral on8