Differentiation revision 1.
Show that the function is never decreasing. 2.
The radius of a circular oil slick is increasing at a rate of 1.5 m/s. Taking π to be 3.14, find to 2 s.f. the rate at which the area of the slick is increasing when its radius is 300m. The radius rcm of a circular ink spot t seconds after it first appears is given by Calculate a) b) c) The time taken for it to double its initial value The rate of increase of the radius r in cm/s when t=3 The value to which r tends as t tends to infinity. 4.
An open rectangular box is made of very thin sheet metal. Its volume is 128 cm 3 , its width is x cm and its length 4x cm. Obtain an expression for its depth in terms of x. Show that the total surface area of its base, its ends and its sides is equal to cm 3 . Calculate the dimensions of the box for which the surface area is a minimum, and explain why your answer gives a minimum rather than a maximum. 5.
Find the coordinates of the turning point on the curve and determine its nature. 6.
Determine the position and nature of the stationary points of the curve , and hence 7.
sketch the curve. A right circular cone with semi-vertical angle θ is inscribed in a circle of radius a, with its vertex and the rim of its base on a surface of the sphere. Prove that its volume is . If a is fixed and θ varies, find the limits within which the volume must lie. 1.
2. 2800 m 2 /s 3.a) 1/3 sec, b)0.28 m/s ; c)4 4.d = 32x -2 , w = 3.42, l = 13.68 cm, depth = 2.74 cm, 5. (-1, 1/3) max, 6. Point of inflexion (0,0), min at (3,-27) 7.0 < V <