An autonomous underwater vehicle (AUV) is descending into a lake. Its depth D D\,D meters relative to its starting point at time t t\,t seconds is modelled by the function D(t)D(t)D(t) for t>0t > 0t>0.
The rate of change of the AUV's depth is given by
D′(t)=3t+4−16t2 D'(t) = 3\sqrt{t} + 4 - \frac{16}{t^2} D′(t)=3t+4−t216At t=4t = 4t=4, the AUV is at a depth of 15 meters, represented by the point P(4,15)P(4, 15)P(4,15) on the curve y=D(t)y = D(t)y=D(t).
Determine the equation of the normal to the curve y=D(t)y = D(t)y=D(t) at the point PPP. Give your answer in the form at+by+c=0at + by + c = 0at+by+c=0, where a,b, a, b,\,a,b, and c c\,c are integers.
Find the expression for D(t)D(t)D(t).
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, 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.
