Mathematics IA Assignment 4 Semester 1, 2016 Submit solutions by 2pm Tuesday April 26th (together with solutions for Assignment 5) Algebra For each of the following say whether the matrix A is invertible or not. You must give an explanation in each case, you can refer to theorems and results from class. 0 1 1 0 2 3 B0 2 4 0C C (a) A = B @1 2 1 3A 1 0 2 3 (b) A is an n n matrix such that the rows of A span Rn . (c) A is an n n matrix that is row equivalent to a matrix B and for every b 2 Rn there exists an x 2 Rn such that Bx = b. Calculus In lectures, we estimated the area under the graph of y = x2 over the interval [0, b] by partitioning [0, b] into n equal subintervals, estimating the area under the curve over each of these subintervals, and adding up the resulting estimates. Then we were able to use a tricky formula to nd the limiting value of these estimates as n became larger and larger. Suppose that we try to do the same thing to nd the length of the curve y = x2 over [0, b]. (a) Given i 2 {1, 2, 3, . . . , n}, write down the endpoints of the i-th interval Ii in a regular partition of [0, b] into n equal subintervals. (b) Let us approximate the curve y = x2 over Ii by a straight line joining the two ends of the curve (over this interval). Write down an expression for the length of this line, simplifying your expression. (c) Now write down an expression involving the symbol that will give the estimate for the length of the curve y = x2 over [0, b]. 1