The curve C C\,C has equation
y=184(2x−k),x≠k2 y = \frac{18}{4(2x - k)}, \quad x \not= \frac{k}{2} y=4(2x−k)18,x=2kwhere k k\,k is a positive constant and k≠2k \neq 2k=2.
Find dydx\displaystyle \frac{dy}{dx}dxdy giving your answer in simplest form in terms of kkk.
The point P P\,P with x x\,x coordinate 1 lies on CCC. Given that the gradient of the curve at P P\,P is -9, find the two possible values of kkk.
Given also that k<2k < 2k<2, find the equation of the normal to C C\,C at PPP, writing your answer in the form ax+by+c=0ax + by + c = 0ax+by+c=0, where a,b a, b\,a,b and c c\,c are integers to be found.
Show, using algebraic integration, that
∫13184(2x−k) dx=λln(5) \int_{1}^{3} \frac{18}{4(2x - k)} \, dx = \lambda \ln(5) ∫134(2x−k)18dx=λln(5)where λ \lambda\,λ is a constant to be found.
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.