Bream Bay College Sample Assessment

Transcription

1 Name: 3 Bream Bay College Sample Assessment Level 3 Mathematics and Statistics (Mathematics) (3.6): Apply differentiation methods in solving problems Credits: Six You should answer ALL parts of ALL questions in this booklet. You should have a copy of the Level 3 Calculus Formula Sheet. Achievement Achievement with Merit Achievement with Excellence Apply differentiation methods in solving problems. Apply differentiation methods, using relational thinking, in solving problems. Apply differentiation methods, using extended abstract thinking, in solving problems. OVERALL LEVEL OF PERFORMANCE New Zealand Qualifications Authority, 2012 All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

2 2 You are advised to spend 60 minutes answering the questions in this booklet. QUESTION ONE (a) Differentiate: y = x 3-2x You do not need to simplify your answer. æπ (b) Find the equation of the tangent to the curve y = tan x at the point ç è 4, 1 ö ø (c) 0.5t A curve is defined by the parameters x = 4e and y = t 2-4t. Find the gradient of the tangent to the curve at t = 6.

3 3 (d) The point P,0 lies on the curve y xsin x Find the equation of the tangent to the curve at point P. (e) A spherical balloon is deflating so that its volume is decreasing at a constant rate of 12 cm 3 per hour. Find the rate at which the surface area of the balloon is decreasing when the volume of the balloon is cm 3.

4 4 QUESTION TWO (a) Differentiate the following: y = e 2 x + π x You do not need to simplify your answer. (b) A car engine is started and is left to idle. The temperature of the water in the car s radiator increases. The temperature of this water is given by the equation: T(m) = 20 m + 5ln(m), 0 < m < 10 where T is the temperature of the water ( C) and m is the number of minutes since the car was turned on. Find the rate at which the temperature of the water is changing 6 minutes after the car engine is turned on.

5 5 (c) Differentiate the following: y = 3x2 + cos x e 4 x You do not need to simplify your answer. (d) A tow truck is using a winch to move an illegally parked car. The winch pulls a rope that runs through a pulley and is tied to the front of the car. The pulley is 2 m higher than the point on the car where the rope is attached. The rope is being pulled through the pulley at a rate of 6 m per minute. The front wheels of the car do not leave the ground. At what rate will the car be moving when there is 8 m of rope between the car and the pulley.

6 6 (e) Find the values of t for which the function defined by x = 2t 3-5 and y = 4t 2-5t is concave up.

7 7 QUESTION THREE (a) Differentiate the following: y = sin( 2x)ln x You do not need to simplify your answer. (b) Find the equation of the tangent to 1 1 y at the point 2, x 2 4

8 8 (c) A 5 m long ladder is placed against a vertical wall. The bottom of the ladder slides out away from the wall, moving at 0.6 m per second. At what rate will the top of the ladder be moving downwards when it is 3 m above the ground? 2x 1 (d) Find the x-value of the point on the curve y 3x e 0 where the gradient is 7.

9 9 (e) A cone has a height of 25 cm and a radius of 10 cm. A cylinder is inscribed in the cone. The base of the cylinder is in the same plane as the base of the cone. The cylinder base has the same centre as the base of the cone. Find the radius of the cylinder so that it has the maximum volume. You may assume that d 2 V dr 2 < 0

10 10 Extra space if required. Clearly number the question (if required). Question number

11 11 Extra space if required. Clearly number the question (if required). Question number