MAS001 SCHOOL OF MATHEMATICS AND STATISTICS Foundation Year Mathematics I Autumn Semester hour 30 minutes ØØ ÑÔØ ÐÐ ÕÙ Ø ÓÒ º Ì ÐÐÓ Ø ÓÒ Ó Ñ

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1 SCHOOL OF MATHEMATICS AND STATISTICS Foundation Year Mathematics I Autumn Semester hour 30 minutes ØØÑÔØ ÐÐ ÕÙ ØÓÒ º Ì ÐÐÓØÓÒ Ó ÑÖ ÓÛÒ Ò ÖØ º ½ ÜÔÖ x x+y + x x y 1 (x+y)(x y) ÒÐ ÖØÓÒ ÑÔÐÝÒ ÝÓÙÖ Ò ÛÖ ÛÖ ÔÓ Ðº ½ ÑÖµ ¾ ËÑÔÐÝ a 2 t 3 x 4 +2a 2 t 3 x 2 y 4 +a 2 t 3 y 8 ta 2 y 8 ta 2 x 4 ÑÙ ÔÓ Ðº ¾ ÑÖ µ ËÑÔÐÝ (10) 2 3 (35) (56) 1 3 ÑÙ ÔÓ Ðº ËÓÛ ÐÐ ÝÓÙÖ ÛÓÖÒ Û ÓÙÐ ÒÐÙ Ù Ò ÐÛ Ó Ò º ¾ ÑÖ µ ÊØÓÒÐÞ Ø ÒÓÑÒØÓÖ Ó º ½ ÑÖµ 1 Turn Over

2 µ ØÓÖÞ 4x 2 7x 2. ½ ÑÖµ µ ËÓÐÚ Ø ÑÙÐØÒÓÙ ÕÙØÓÒ { 3x 2 6y x = 20 x 2 y x+3 = 0. ÑÖ µ µ ÓÑÔÐØ Ø ÕÙÖ ÓÖ x 2 +6x+8. ½ ÑÖµ µ ËÓÐÚ x 2 +6x+8 = 0. ¾ ÑÖ µ µ Úµ ÖÛ Ø ÖÔ Ó y = x 2 +6x+8 Ò ÒØ ÐÖÐÝ Ø ÑÒÑÙÑ ÔÓÒØ Ó Ø ÙÖÚ Ò ÛÖ Ø ÙÖÚ ÖÓ Ø x¹ Ò y¹ü º ¾ ÑÖ µ ÄØ f Ø ÙÒØÓÒ ÛØ ÓÑÒ {x R : 6 < x 1} Ò ÖÙÐ f(x) = x 2 +6x+8º Ò Ø ÖÒ Ó fº ½ ÑÖµ ÄØ p(x) = x 4 7x 2, q(x) = 4x 3 5 Ò r(x) = 2x 2 +2x+1. µ ÜÔÒ p(x) q(x)r(x), ÓÐÐØÒ Ð ØÖÑ ØÓØÖº ÏØ Ø Ó¹ ÒØ Ó x 3 Ò p(x) q(x)r(x) ¾ ÑÖ µ µ Ò r(q(x))º ¾ ÑÖ µ ÜÔÖ Ø ÓÐÐÓÛÒ ÖØÓÒ ÔÖØÐ ÖØÓÒ º ÓÙÖ Ò ÛÖ ÓÙÐ ÒÐÙ º 4x 11 µ (x+4)(2x 1). ÑÖ µ µ 2x 5 8x 3 +5x 2 4. x 2 (x 2 ÑÖ µ 4) 2 Continued

3 ÄØ f Ø ÙÒØÓÒ ÛØ ÖÙÐ f(x) = 1+ x 2 x Ò ÓÑÒ {x R : x 2}º µ Ò Ø ÒÚÖ ÙÒØÓÒ f 1 º ÑÖ µ µ ÏØ Ø ÓÑÒ Ó f 1 ½ ÑÖµ µ ÏØ Ø ÖÒ Ó f 1 ½ ÑÖµ ½¼ ËØ Ø ÖÔ Ó Ø ÓÐÐÓÛÒ ÙÒØÓÒ ØÛÒ π Ò π ÑÒ ÙÖ ØÓ ÑÖ ÑÔÓÖØÒØ ÔÓÒØ º µ y = tan(x)º ½ ÑÖµ µ y = tan(x) 1º ¾ ÑÖ µ µ y = tan(x)º ½ ÑÖµ ½½ ËÑÔÐÝ ÓÙ ÓÙÐ ÓÛ ÐÐ ÝÓÙÖ ÛÓÖÒº ln(e x+2 +2e x+1 +e x ) x. ÑÖ µ ½¾ ËÓÐÚ Ø ÓÐÐÓÛÒ ÕÙØÓÒ ÓÖ x ÚÒ ÝÓÙÖ Ò ÛÖ Ò ØÖÑ Ó ln2 Ò ln5º ÓÙÖ Ò ÛÖ ÑÙ Ø ÜØ Ó Ó ÒÓØ Ù ÐÙÐØÓÖº 5 x 1 = (4e)x 10 x+1. ÑÖ µ End of Question Paper 3

4 SCHOOL OF MATHEMATICS AND STATISTICS Mathematics I Spring Semester hours Attempt all the questions. The allocation of marks is shown in brackets. 1 Dierentiate the following functions with respect to x: (i) f(x) = 3x x 1 ; f(x) = (sin x).(ln x); (iii) f(x) = ex cos x ; (iv) f(x) = e sin x. (8 marks) 2 (i) State the denition of the derivative of a function f(x) (your denition should involve a limit). (3 marks) Use the method of rst principles to dierentiate f(x) = x 1. (5 marks) 3 (i) Find all stationary points on the curve described by the graph of the function f(x) = x6 6 x4, and determine their nature. (8 marks) Use the information from Q3 (i) to sketch the graph of f(x). (4 marks) 4 (i) Find dy dx when x(t) = e2t and y(t) = tan t. (4 marks) Find dy dx when xy + 3y + x2 = 0. (4 marks) (iii) Find dy dx when y = 3x. dierentiation). (Hint: take natural logarithms and use implicit (4 marks) 1 Turn Over

5 5 (i) You are given that the surface area of a sphere of radius r is 4πr 2 and that the radius of the sphere is decreasing at a rate of 1mm per second. Calculate the rate of change of the surface area when the radius of the sphere is 1cm. (5 marks) Determine the maximum area of a rectangle with xed perimeter 8cm. (5 marks) 6 Find the indenite integrals in Parts (i) and, and evaluate the denite integrals in Parts (iii) and (iv). You should simplify your answers as much as possible. (i) 3x 2 x 2 dx; xe x2 dx. (iii) (iv) 2 1 ln 3 ln 2 1 x dx; xe x dx. (10 marks) 7 Dierentiate f(x) = sin 1 ( x 3 ) with respect to x. x Hence or otherwise nd dx. 1 x3 (8 marks) 8 Find the following indenite integrals: (i) (sin(x)) 3 dx; sin( x) x dx. (10 marks) 9 Let f(x) = x 2 2x 3 and g(x) = (x + 1)(x 2 4). By simplifying f(x) 4 3 f(x) g(x) dx. g(x) evaluate (10 marks) 2 Continued

6 10 (i) Determine the area of the region bounded by the lines x = 1, x = 1 and the graphs of the functions f(x) = x 3 and g(x) = x 3. (Remark: the region looks a fancy bowtie ). Using the fact that the semi-circle in the upper half plane centred at (0, 0) of radius r is the graph of the function f(x) = + r 2 x 2, nd the volume of the sphere of radius r. (12 marks) End of Question Paper 3