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4. For each of the following if the statement is true, briefly justify why. If it can fail, provide an example to show how it can fail. a) Any open cover of (0,3) will contain a finite subcover for (1,21. b) If f continuous on (0,3) then f is uniformly continuous on (0,3). c) If f is an antiderivative for some function on 1-1,11 then f is bounded on 1-1,1). d) If f is continuous on [1,3) then there is a function with F'(x) = f(x). e) If {ak}no converges and f(x) is increasing and bounded then f(ak) converges. g) If f attains a maximum and minimum on 1,2] then f is integrable on (1,2). f) If Dk=0 ak converges then f(x) = [wo-o ak is a continuous function. La 1 7. Then g(f(x)) is integrable on [a,b]. Theorem: Let *(x) be integrable on la,b) and suppose g(x) is differentiable with 19'(x)l < K for 3 E la,bl. ( In our usual notation let M! - sup {j(x) I € € (21-1,211) and m² - inf (J(x) IXE 101-1,211) And similarly M?os = sup {g(f(x)) | x E (x1-1,Fil) and mi°or = inf {g(f(x)) \ a) Consider subinterval [x1-1,x1) Show for z E [11-1,3 that \g(f(z)) - 9(f(24-1111 b) Use this to show that M?°f g < +9(f(21-1) and that m?os (Hint:Apply the MVT on on the interval Conclude that MPOS -mmoo 2K(M; - m(). c) Use the fact that f is integrable on [a,b) to find a partition of la,t b] so that E(MPO AAI, Conclude the theorem. which satisfied 8. Let f(x) be a continuous and strictly monotonic increasing function defined on (1,00) which satisfies the property f(x1x2) = f(x1) + f (x2) for all x1,T2 E (1,00). a) Prove that f(1) = 0 and that f (x) maps [1, 00) onto the entire interval [0, 00). b) Show f(x) is 1-1 and hence the inverse f-1(y) is well defined for all points in [0,00). Then prove that f (y) satisfies the property for all 1 11,32 E/0,00). c) Now let g(x) be any function that satisfies the properties of the inverse function found in part (b). That is, g(x1 + x2) = g(x1)g(x2) and g(0) = 1. Prove that if g(x) is differentiable at x = 0 then it must be differentiable everywhere with derivative g' (a) = g'(0)g(a). What does this imply about the derivative of f(x) in part (a)? (Hint: Use the f'(a) = limh-+o f(ath)-/(a) definition for the derivative. Use the property proved class that if f(x) is differentiable at x = a then its inverse will be differentiable at 6 = f(a) and (f-¹)(b) = )

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