(i) The cost function CCC, in thousands of pounds, for a specialized manufacturing process is modeled by
C(x)=(2x+5)23x−1,x≠13 C(x) = \frac{(2x + 5)^2}{3x - 1}, \quad x \neq \frac{1}{3} C(x)=3x−1(2x+5)2,x=31where x x\,x represents the amount of catalyst used in kilograms.
Find C′(x)C'(x)C′(x) in the form P(x)Q(x)\displaystyle \frac{P(x)}{Q(x)}Q(x)P(x) where P(x)P(x)P(x) and Q(x)Q(x)Q(x) are fully factorised quadratic expressions.
Determine the range of values of x x\,x for which the cost function C(x)C(x)C(x) is increasing.
The vertical displacement h h\,h of a floating buoy, in metres, is modeled by the function
h(t)=tsin6t,0≤t<π6 h(t) = t\sqrt{\sin 6t}, \quad 0 \le t < \frac{\pi}{6} h(t)=tsin6t,0≤t<6πwhere t t\,t is the time in seconds after a wave passes. The buoy reaches its maximum height at a point MMM.
Show that the ttt-coordinate of M M\,M satisfies the equation tan6t+kt=0\tan 6t + kt = 0tan6t+kt=0, where k k\,k is a constant to be found.
Practise Edexcel A Level Maths 9.5 The Quotient Rule with exam-style questions for A Level Maths. 29 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.