The rate of change of the volume of water in a large industrial storage tank, R(t)R(t)R(t) in m3/day\text{m}^3/\text{day}m3/day, is modeled by the function:
R(t)=54t2+2t−14,t>0 R(t) = \frac{54}{t^2} + 2t - 14, \quad t > 0 R(t)=t254+2t−14,t>0where t t\,t is the time in days since the start of a maintenance cycle. Using calculus:
Find the set of values of t t\,t for which the rate of change R(t)R(t)R(t) is increasing, giving your answer in the form t>ab3t > a\sqrt[3]{b}t>a3b where a a\,a and b b\,b are integers.
Show that ∫39(54t2+2t−14)dt=0\displaystyle \int_{3}^{9} \left( \frac{54}{t^2} + 2t - 14 \right) dt = 0∫39(t254+2t−14)dt=0.
Given that ∫36(54t2+2t−14)dt=−6\displaystyle \int_{3}^{6} \left( \frac{54}{t^2} + 2t - 14 \right) dt = -6∫36(t254+2t−14)dt=−6:
(i) State the value of ∫69(54t2+2t−14)dt\displaystyle \int_{6}^{9} \left( \frac{54}{t^2} + 2t - 14 \right) dt∫69(t254+2t−14)dt.
(ii) Find the value of the constant k k\,k such that ∫36(54t2+2t+k)dt=0\displaystyle \int_{3}^{6} \left( \frac{54}{t^2} + 2t + k \right) dt = 0∫36(t254+2t+k)dt=0.
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.