2. Prove that a metric space X is discrete if and only if every function on X to an arbitrary second metric space is continuous.

3. Let X be a dense subset of a metric space Y. Suppose that every Cauchy sequence in X converges to a point in Y. Prove that Y is complete.

1. Explain why the Comparison Test can or cannot be used to decide if the series converges or diverges.

2. Use the Comparison Test or Limit Comparison Test to determine whether each series is convergent or divergent. Carefully explain your justification for using each test.

3. Explain why the Integral Test can or cannot be used to determine whether the series is convergent.

Let z= {x ε r: f(x)=0} f is a continuous function. Show that

2. For the functions f and g defined by, f(x) = cosh (3x), g(x) = log(x) find the composite functions fog and go f stating clearly the intended domain and range.

4. Determine whether the following series converge or diverge. In the case of convergence, find the limit.

Determine whether each series is convergent or divergent. For each alternating series that is convergent, determine whether it is conditionally or absolutely convergent.

(a) Draw and shade in the rectangles needed to represent a left sum approximation of the distance traveled on the interval [0,30] with n = 6 rectangles. (b) Use the graph to estimate the distance traveled on the interval [0,30] with a left Riemann sum with n = 6 rectangles. Note: 1 second is equal to 1/3600 hours. (c) Does this give an overestimate or an underestimate of the distance traveled? (d) Interpret the Riemann Sum in the context of the situation. Include units in your answer.

Sketch the rectangles for the specified Riemann Sums to estimate the area under the curve of f on the interval [2, 8].

(3) OpenStax Section 1.1 435. Let r; denote the total rainfall in Portland on the jth day of the year in 2009. Interpret the following in the context of the situation.

(a) Use the table to determine a midpoint Riemann Sum with n = (b) Interpret your answer in the context of that situation. What are you estimating with this Riemann Sum?

Determine inf{a,: n N), sup{a,: n N}, lim,Oinfa, and limn 00 supan:

(Squeeze theorem, sandwich theorem, squeeze lemma) (a) Let (an), (bn), (cn) be real consequences with

Let A 1(a) Be (G,-) a group. For which x € G does x2 = x apply? (b) Specify three different o € S4 for which 03 = o applies. How many o € S4 are there any that satisfy this equation? (1 + 3 Points)