We define a subring of a ring in the same way we defined a subgroup of agroup: is a subring of if and only if is a ring,and is a ring with the same operations. For example, the ring of evenintegers is a subring of the ring of integers, and both are subrings of the ringof rational numbers.(a) Prove that the ring is a subring of any ring (calledthe trivial subring).(b) (Subring Test) Prove that if is a ring, T is a nonempty subset ofR, and T is closed under subtraction and multiplication, then isa subring.

S343 Section 2.7 Notes- Euler’s Method of Numerical Approximation 9-13-16 Used for equations that cannot be solved without computer- give numerical approximation of at given values Known as first-order method/tangent line method- error at each step (iteration) decreases like ℎ decreases +ℎ − ) Recall = (, )) = ℎ→0 ( ℎ ) where ℎ = uniform step size between and +1 (ie. = + ℎ) +1 +ℎ −() o (, )) ≈ ℎ ℎ( (, ))≈ + ℎ − ( )