Mathematics for Analytics and Finance Assignment 1 Submit by March 12, 2024, 11:59PM Beijing Time. Let f be the function defined by y = f(x) &= v is the largest solution of y? = 3x% 2xy a) Find a formula for . (5 marks) b) What are the domain and range of 7 (5 marks) . A function f is given which satisfies 1 f( ):2x12 x+1 for all real numbers x. Compute (5 marks): a) f(1) b) f(0) ) f(x) d) f(v e) f(f(2)) where x and t are arbitrary real numbers. What are the range and domain of 7 (5 marks) Find the derivatives of the following functions (10 marks): a) x2 o) =t b) ax+ b lc(x)_cx+d ) 1+ x2 ) =757+ () = 1 _2x4+7 g\\ Cf(x) 14 x? d) X f(x) = () 14+ x 4. For each of the following problems find the derivative f(x) ify = f(x) satisfies the given equation (10 marks): a) Xy = b) sin (xy) = c) xy xty =1 e (y + 1)2+ y- x = 0 e) x3 + xyty3 =3 5. Compute those derivatives (10 marks): a) Jo ( 1 + + 2 ) at b) 1 Itxzdx c) at e2x dx d) d 2x s'ds e) d 1 dt Jo 1+ xz ax6. Compute the following indefinite integrals (10 marks): a) (6 x5 - 2x- 4 - 7x}dx b 16-5+4ex + 78 , dx c) 1 ( 2* + (2 ) }dx d) S ( + " + x a + a* + ax) dx 7. Compute the following definite integrals (10 marks): a) b C [ ( x + 4 )"ax d) In6 8exdx In3 8. Find the first 4 terms of the Taylor series for the following functions (10 marks): a) In x centered at a=1 b) 1/x centered at a=1 c) sin x centered at a = " . 9. Find the first 3 terms of the Taylor series for the function sin wx centered at a=0.5. Use your answer to find an approximate value to sin (? + To ) (10 marks) 10. Find the first 4 terms in the Taylor series for (x - 1)e* near x=1 (10 marks)