Question

#55, 59

In Exercises 55 and 56, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. 55. If f is continuous at x = a, then f is differentiable at x = a. 56. If f is continuous at x = a and g is differentiable at x = a, then lim f(x)g(x) = f(a)g(a). X 57. Sketch the graph of the function f(x) = x + 1), and show that the function does not have a derivative at x = -1. 58. Sketch the graph of the function f(x) = 1/(x - 1), and show that the function does not have a derivative at X = 1. 59. Let f(x) = { fx? Tax + b if x si if x > 1 Find the values of a and b such that f is continuous and has a derivative at x = 1. Sketch the graph of f.

Answer 1

And wer 55 В f is continuous at nea, thin finmay or may not be differentia ble at uca Exameli- f(u) = 14/= {4, no. Hose f(n) is continuous at uzo but not differentia ble at noo Answer 52 if ul1 is use a f is continuous and has dirivative tu= s k² laut b find a fb such that at u= 1. Y f is continuous at lim tlula lim n=1 flu) = then f(1) ni nagt lim antb = (1) ? lim f(1) fui +12 ni 1 - lim a (1-h) + 6 = ho a+b=1 - 0 differentiable at u=1 then If t is lim Hul - HK) - lim flu)-f(K) not nk nk nk lim n²-(1)2 = lim auto (ath) à u gt n-1 n1 - 1

lim hth (1-14u+1) - lim nur UI aut b arb nog - lim (1+h+1) - ho - 2 = Lim lim a (1-6).- a no 1-6-1 &-ahab th a=2 - to lim ho a a= 2 by 0 4 b=-1 fluga s 42 then { 2U-1 is n=1 is u71 flulau2 24-1=fln) 100 U 1