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10 Nov 2019
(2) Observe from parts (b) and (c) in the previous problem that This may lead you to believe the following for a general function g(x): The left hand limit of g'(x) at e is equal to the left hand derivative of g(x) at e and the right hand limit of g'(x) at e is equal to the right hand derivative of g(x) at c. In general, this will not always be the case. However, for the purposes of this course, it will be the case as long as g(x) is continuous at e. (a) Using the previous remark, show that the function is differentiable at 0 by first showing it is continuous there, and then calculating by using the remark preceding Question (1). (b) Show that continuity is required to apply the result by calculating bothfor the function and showing they are not equal (in fact the latter doesn?t even exist). As a sidenote, observe in this example that lim x tends to 1- f'(x) = lim x tends to 1+ f'(x) but f'(1) does not exist because f is not continuous at 1.
(2) Observe from parts (b) and (c) in the previous problem that This may lead you to believe the following for a general function g(x): The left hand limit of g'(x) at e is equal to the left hand derivative of g(x) at e and the right hand limit of g'(x) at e is equal to the right hand derivative of g(x) at c. In general, this will not always be the case. However, for the purposes of this course, it will be the case as long as g(x) is continuous at e. (a) Using the previous remark, show that the function is differentiable at 0 by first showing it is continuous there, and then calculating by using the remark preceding Question (1). (b) Show that continuity is required to apply the result by calculating bothfor the function and showing they are not equal (in fact the latter doesn?t even exist). As a sidenote, observe in this example that lim x tends to 1- f'(x) = lim x tends to 1+ f'(x) but f'(1) does not exist because f is not continuous at 1.
Trinidad TremblayLv2
19 Sep 2019