MAT1512 ExAM PAck 2024 LATEST QUESTIONS AND CORRECT ANSWERS

a) Determine the following limits (if they exist): (i) 3 1 6 2 lim 2      x x x (3) (ii) 3 9 lim 2 3      x x x (3) (iii) x x x x x 2 4 2 lim 2     (3) (iv) xx x    1 1 lim 1 (3) (v) 3 9 2 lim  0  x  x x (3) (vi) t t t t sin tan 2 lim 0   (3) (b) Use the Squeeze Theorem to determine the following limit: . 10 5 cos3 lim 2 2     k k k k (3) (c) Let          2 2 2 2 ( ) x if x x if x G x (i) Draw the graph of Gx. (1) (ii) Determine lim ( ) 2 G x x  . (2) (iii) Is Gx continuous at x  2 ? Give a reason for your answer. (1) [25] QUESTION 2 (a) By the first principles of differentiation, find the derivative of f x 2x2  3x  4 at x  2. (5) (b) Find the derivatives of the following functions by using the appropriate rules of differentiation: (i)   x x x x f x sin cos 2   (3) (ii) gx cos5xsinx2  (3)3 MAT1512 January /February 2022 (iii)     3 sin 3 z z F z xdx (5) (c) Given: sinx  y 2x , find the following: (i) dx dy by using implicit differentiation. (4) (ii) the equations of the tangent line and normal line to the curve sinx  y 2x at the point 0, .