In a bio-reactor, the rate of oxygen consumption V(t)V(t)V(t), in milligrams per hour, is modelled by the equation
V(t)=(10t+13)2t+3 V(t) = (10t + 13)\sqrt{2t + 3} V(t)=(10t+13)2t+3where t t\,t is the time in hours since the start of an experiment, 0≤t≤30 \le t \le 30≤t≤3.
Use the substitution u=2t+3u = 2t + 3u=2t+3 to show that
∫03(10t+13)2t+3 dt \int_{0}^{3} (10t + 13)\sqrt{2t + 3} \, dt ∫03(10t+13)2t+3dtcan be written as
12∫39(5u−2u0)u12 du=12∫a9(5u32−2u12) du \frac{1}{2} \int_{3}^{9} (5u - 2u^0)u^{\frac{1}{2}} \, du = \frac{1}{2} \int_{a}^{9} (5u^{\frac{3}{2}} - 2u^{\frac{1}{2}}) \, du 21∫39(5u−2u0)u21du=21∫a9(5u23−2u21)duwhere a a\,a is a constant to be found.
Hence, or otherwise, show that the total oxygen consumed over the 3-hour period is
225−73 mg 225 - 7\sqrt{3} \text{ mg} 225−73 mgA scientist uses three rectangles of equal width to approximate the total oxygen consumed, VtotalV_{total}Vtotal, using the left-hand edge method. The total area of these three rectangles is RRR.
The scientist decides to improve the approximation by increasing the number of rectangles used (still using the left-hand edge method).
Explain why the value of this improved approximation will be greater than RRR, but less than 225−73225 - 7\sqrt{3}225−73.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.