Question 1 - 15 marks You should be able to answer this question after studying Unit 7. Differentiate the following functions, simplifying your answers as far as possible. (a) f(z) = x?sin(2x) 3 (b) g(y) = tan(y)yl sec(y) (c) h(z)= (e' +2%)* (d) k(t) =sin(t2 + 1) cos(t? 1) Question 2 - 10 marks You should be able to answer this question after studying Unit 7. An ornamental container consists of a cuboid-shaped hole inside a sphere. The part of the sphere above the cuboid is the lid of the container. The diagrams show a vertical cross section of the container through a face of the cuboid, and on the following page, a three dimensional representation of the situation. Cuboid-shaped Lid of the container hole y cm ycm X cm X cm +> X cm Vertical cross section through a face of the cuboid The cuboid removed The cuboid has height y cm and a square cross-section, with the side of the square of length x cm. The radius of the sphere is Rcm. (a) Show that z2 = 2R% 142 3] (b) Show that the total volume of the cuboid hole V' m3 , in terms of y is given by V = 2R%y %y3. (2] (c) Starting with the equation in (b), use differentiation to determine the height y that gives the maximum volume V for the cuboid, and show that this is a maximum. What is the maximum volume? (Give both answers in terms of R.) [5] page 3 of 11 The diagram below is a three dimensional representation of the container in question 2. The dashed lines indicated on the diagram meet at the centre of the sphere. Question 3 - 5 marks You should be able to answer this question after studying Unit 7. Find the indefinite integral of the function f(2) = (25 - 4) (3z + 2) (Hint: It may be helpful to expand and simplify f(z) first.) [5]