MATH117 Midterm: MATH117 Exam Midterm 3 2014 Fall Solution
Document Summary
Math 117 fall 2014 midterm exam 3 solu- tions. please write clearly and show enough work. Math 117 fall 2014 midterm exam 3 solutions a =/ 0 but discontinuous at 0. Question 1. (5 pts) prove by " : f (x) := 2 x > 0. Then for every jx aj < , we have either both x; a > 0 or both x; a < 0. Consequently (1) jf (x) f (a)j = 0 < " is continuous at every. At 0, let > 0 be arbitrary. Then we have and discontinuity follows. jf (x) f (0)j = 1 (2) Question 2. (5 pts) let f (x) :=( x + x2 sin 1. Prove that f is dier- entiable everywhere on r and calculate f 0(x). Since x; x2; sin x are dierentiable everywhere and 1/x is dierentiable everywhere except at 0, we have x + x2 sin 1 x dierentiable at every x =/ 0.
