(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.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.