2. (a) Find the Maclaurin polynomials of degree 1 for sin(:1:) and ln(1 + as) and then write these functions using their degree 1 polynomials and expressing the remainder using Big O notation. (b) Use the expressions from part (a) to evaluate the following limit. lim :1: + Sln(a:) ln(1 + at) :L'>0 :1: 3. (a) Find the Maclaurin polynomial 12201:) for f (x) = \\/1 cc and then write this function using p2(a:) and expressing the remainder using Big 0 notation. (b) Use the expression from part (a) to evaluate , 2V1 I:+a:2 11m . 90)0 122 4. Use Maclaurin polynomials with Big 0 notation to evaluate the following limits. (a) lim m (b) 11m M (c) lim 003(2-76) (1 + M2 w>0 m2 50)0 (1:2 x>0 a: sinm _ , sinm for :1: y 0 and smc(0) 2 11m 113 m)0 13 =1. 5. The sinc function is defined as sinc(:1:) = This function appears frequently in signal processing applications. (a) Using Python or Desmos, graph the sine function on [27r, 27r]. (b) We would like to estimate the :1: values of the relative minimum points on this graph. Note that these are symmetric since sinc(:c) is an even function. Using optimisation and New- ton's method with suitable initial values, find these a: values accurate to 6 decimal places. It will help to simplify the derivative function as much as possible first before implementing Newton's method. (0) Optional: Use Python to sketch a convergence plot showing iteration number vs current approximation. 6. Consider the matrix A = [3 12)]- (a) Sketch the unit square and its image under the transformation A. (b) What type of transformation does A represent? (c) What is the signed scaling factor involved in this transformation, that is, what is det(A)? (d) If a matrix B is obtained from A by the elementary column operation 01 02, what is det B? 7. Consider the matrix A = [g (1)]. (i) Sketch the unit square and its image under the transformation A. (ii) Describe the transformation and find the determinant of A. 8. Find the determinants of the following 2 x 2 matrices and state whether they are invertible or not. How is the determinant related to invertibility? O O (a) A = OH' (b) B = (c) C = OH 9. If det A = 3 and det B = -6, find the following determinants. (a) det( AB) (b) det(A-1) (c) det(AB-1)