Calculus Questions

1. b Evaluate the following limits: I2 2 x²-2 i) lim ii) lim iii - z-+1* x 1 x z-++00 I' 2 I-2 iv lim v lim vi lim - z-++00 3x³ 1 1->-00 3x³ 1 r->-00 I' 2 I 2 x²-3x-4 vii lim vil lim ix z-++00 3.5 1 z->-00 3r¹ 1 z-+4 x-4 lim 2. (c) Find a value of K so that the function f(x) is continuous at x = 0 where sinz + K(x-3)², <0, f(x) x - - 3. @ - = - Calculate lim Af A-+0 h by Evaluate lim tan(2x) r-+0 sin(4x) C Evaluate lim z-++00 b Use the definition of derivative to find j'(2) for f(x) = I2 - 3x. c Find the equation of the tangent line of the graph of the function f(x) = x² - 3x at the point (2,-2). What is the area of the triangle formed by this tangent line and the co-ordinate axes? 5. Find the equation of the tangent line to the curve y - 2r³ - 3x2 - 3Gx - 82 at its point of inflection (b) Let y(x) be defined implicitly by the equation 1y - r+ 2y. Find y at the point (2,1). 6. Find the derivatives of the following functions Simplify your answers go arctanz c In |sec r + tan x) cot: d) aresin r + arccos I (x 1)2 7. Let /(x) = In|(x - 1)(x + 1)(x - 3)]. What is the domain of the function f(x) ? For what values of x in the domain of 8 Prove that 9. b Show that the equation f(x) = 2x³-3x²-36x-82 - 0 has a solution for some x* € (4,5). c Apply Newton's method to this equation with an initial guess To - 4 to find T1 and evaluate /(I)). Now use To = 5 to find x1 and evaluate f(x1) for this T1. Which one of these two approximations would you use for x*? 10 Use 'Hopital's rule to find the following limits: 1-e-2 a z-+0 In(1+1) lim tan: b linn =-+0 x -sin 11 Use the 7 step method to sketch the graph of the curve y = 2r³ - 3x2 - 36x - 82. 12. b) Find a value of c that satisfies the conclusion of the MVT for f(x) = +2 - x - 6 on the interval (1,4).