An experimental physicist is using the trapezium rule to estimate the total impulse (the area under a force-time graph) for four different test engines over the interval 0≤t≤50 \le t \le 50≤t≤5. The force F(t)F(t)F(t) (in Newtons) for each engine is modeled by the following functions:
Engine A: F(t)=1.5t+20F(t) = 1.5t + 20F(t)=1.5t+20 Engine B: F(t)=0.4t2+10F(t) = 0.4t^2 + 10F(t)=0.4t2+10 Engine C: F(t)=6t+4F(t) = 6\sqrt{t+4}F(t)=6t+4 Engine D: F(t)=60t+3\displaystyle F(t) = \frac{60}{t+3}F(t)=t+360
Identify the engine model for which the trapezium rule will produce an underestimate of the actual impulse over this interval. Justify your answer by considering the concavity of the models.
125 exam-style questions on OCR A Level Maths 1.9 Numerical Methods (A-level only), covering 1.9.1 Locating roots by sign change (A-level only), 1.9.2 Failure of sign change methods (A-level only), 1.9.3 Simple iterative methods (A-level only), 1.9.4 Newton-Raphson method (A-level only), 1.9.5 Failure of iterative methods (A-level only), 1.9.6 Numerical integration (A-level only), and 1.9.7 Numerical methods in context (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.