Let A be a 3 x 5 matrix. (a) Give a/l possible values for the rank of A. (b) If the rank of A is 3. what is the dimension of its column space? (c) If the mnk of A is 3. what is the dimension of the solution space of the homogeneous system Ax = 0'1

Math 1314 Lesson 9: Marginal Analysis Marginal Cost Suppose a business owner is operating a plant that manufactures a certain product at a known level. Sometimes the business owner will want to know how much it costs to produce one more unit of this product. The cost of producing this additional item is called the marginal cost. Example 1: Suppose the total cost in dollars per week by ABC Corporation for producing its best-selling product is given by(x) 10,000 3000x 0.4x .2 Find the actual cost of st producing the 101 item. First of all, if you calculate C(101), this is the actual cost of producing 101 items. Similarly, C(100) is the actual cost of producing 10items. So the cost of producing the 101 item can be found by computing the average rate of change, Cx))(Cx C (101) C (100) that is, by computing () hx = 101100 where x = 100 and h = 1. This will give us the actual cost of producing the 101 item. However, it is often inconvenient to use. For this reason, marginal cost is usually approximated by the instantaneous rate of change of the total cost function evaluated at the specific point of interest. That is to say, we’ll find the derivative and substitute in our point of interest. Example 2: Suppose the total cost in dollars per week by ABC Corporation for producing its 2 best-selling product is given by ( ) 10,000 3000 x 0.4 . Find C'(100) a