The depth, hhh metres, of an underwater research drone exploring a lake is modelled by a function h(t)h(t)h(t), where t>0t > 0t>0 is the time in seconds after it passes a specific underwater marker.
The vertical acceleration of the drone is given by the equation
h′′(t)=10t3+12t2 h''(t) = \frac{10}{\sqrt{t^3}} + 12t^2 h′′(t)=t310+12t2A point P(1,5)P(1, 5)P(1,5) lies on the depth-time curve.
Given that the rate of change of depth h′(t)=−2h'(t) = -2h′(t)=−2 at point PPP,
find the equation of the normal to the curve at PPP, writing your answer in the form h=mt+ch = mt + ch=mt+c, where mmm and ccc are constants,
determine an expression for h(t)h(t)h(t).
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.