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5. 6 pt Determine whether the function f(x) is continuous and/or differentiable at x = 1. (x2+1 f(x) = { 12, >1 1 <1
(4) Let f(x) (0 if x<0 (a) Show that f is differentiable at z (b) Is f'continuous on R? Is f continuous on R? Justify your answer.
Use the Mean Value Theorem to demonstrate that In(1 + x) < x, given that x > 0.
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.
Let f(x) be an integrable function such that 1 -4 -4 and 1. 2 Compute the following integrals. Number => [f(e) de » L, (+37(e) +2) de Number c) Se f(a) de = Number
this curve, with X - axis rotation, calculate surfacea area y=e-72, (x > 0)
Problem 2. Let f be a self-map on a set X. For x,y e X define x ~ y if and only if f"(x) = f(y) for some integers n, m > 0. Show that ~ is an equivalence relation.
Exercise 6. Show that if f(x) > 0 for all x e [a, b] and f is integrable, then Sfdx > 0.
Q1 Question 1 2 Points Find the inverse of f-1 of the function f(x) =1+1, 2 > 0. of'() = -1 of '(x) = -1 of '(x) = Of-l does not exist
8. Let f (x) e, 0 > 0; x> 0 (1 1 +e (a) Show that f(x) is a probability density function (b) Find P(X> x) (c) Find the failure rate function of X