A structural engineer models the rate of change of deflection y y\,y of a loaded beam using the gradient function:
dydx=7x+1(x−2)(2x+1)2,x>2 \frac{dy}{dx} = \frac{7x+1}{(x-2)(2x+1)^2}, \quad x > 2 dxdy=(x−2)(2x+1)27x+1,x>2Find the values of the constants AAA, B B\,B and C C\,C such that
7x+1(x−2)(2x+1)2≡Ax−2+B2x+1+C(2x+1)2 \frac{7x+1}{(x-2)(2x+1)^2} \equiv \frac{A}{x-2} + \frac{B}{2x+1} + \frac{C}{(2x+1)^2} (x−2)(2x+1)27x+1≡x−2A+2x+1B+(2x+1)2CHence find the exact change in deflection between x=3x=3x=3 and x=4x=4x=4 by calculating
∫347x+1(x−2)(2x+1)2 dx \int_{3}^{4} \frac{7x+1}{(x-2)(2x+1)^2} \, \mathrm{d}x ∫34(x−2)(2x+1)27x+1dxgiving your answer in the form plnq+rp \ln q + rplnq+r where ppp, q q\,q and r r\,r are rational numbers.
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.