(i) The cost function CCC, in thousands of pounds, for a specialized manufacturing process is modeled by
C(x)=(2x+5)23x−1,x≠13 C(x) = \frac{(2x + 5)^2}{3x - 1}, \quad x \neq \frac{1}{3} C(x)=3x−1(2x+5)2,x=31where x x\,x represents the amount of catalyst used in kilograms.
Find C′(x)C'(x)C′(x) in the form P(x)Q(x)\displaystyle \frac{P(x)}{Q(x)}Q(x)P(x) where P(x)P(x)P(x) and Q(x)Q(x)Q(x) are fully factorised quadratic expressions.
Determine the range of values of x x\,x for which the cost function C(x)C(x)C(x) is increasing.
The vertical displacement h h\,h of a floating buoy, in metres, is modeled by the function
h(t)=tsin6t,0≤t<π6 h(t) = t\sqrt{\sin 6t}, \quad 0 \le t < \frac{\pi}{6} h(t)=tsin6t,0≤t<6πwhere t t\,t is the time in seconds after a wave passes. The buoy reaches its maximum height at a point MMM.
Show that the ttt-coordinate of M M\,M satisfies the equation tan6t+kt=0\tan 6t + kt = 0tan6t+kt=0, where k k\,k 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.