#9.1
Know and use that the gradient of a curve at a point is given by the gradient of the tangent at the point.
#9.2
Know and use that the gradient of the tangent at a point A on a curve is given by the limit of the gradient of chord AP as P approaches A along the curve.
[Excludes: The modulus function.]
#9.3
Understand and use the derivative of \(f(x)\) as the gradient of the tangent to the graph of \(y = f(x)\) at a general point \((x, y)\).
Know that the gradient function \(\dfrac{dy}{dx}\) gives the gradient of the curve and measures the rate of change of \(y\) with respect to \(x\).
Be able to deduce the units of rate of change for graphs modelling real situations. The term derivative of a function.
Notation:
\(\dfrac{dy}{dx} = \lim\limits_{δx→0} \dfrac{δy}{δx}\)
\(f'(x) = \lim\limits_{h→0} \Bigg(\dfrac{f(x+h) - f(x)}{h} \Bigg)\)
#9.5
Be able to differentiate \(y = kx^n\) where \(k\) is a constant and \(n\) is rational, including related sums and differences.
Differentiation from first principles for small positive integer powers.
#9.6
Understand and use the second derivative as the rate of change of gradient.
Notation:
\(f''(x) = \dfrac{d^2y}{dx^2} \)
#9.7
Be able to use differentiation to find stationary points on a curve: maxima and minima.
Distinguish between maximum and minimum turning points.
#9.8
Understand the terms increasing function and decreasing function and be able to find where the function is increasing or decreasing.
In relation to the sign of \(\dfrac{dy}{dx}\).
#9.9
Be able to find the equation of the tangent and normal at a point on a curve.
#9.10
Be able to differentiate \(e^{kx}\), \(a^{kx}\) and \(\ln{x}\).
Including related sums, differences and constant multiples.
#9.11
Be able to differentiate the trigonometrical functions: \(\sin{kx}\); \(\cos{kx}\); \(\tan{kx}\) for \(x\) in radians.
Including their constant multiples, sums and differences. Differentiation from first principles for \(\sin{x}\) and \(\cos{x}\).
#9.12
Be able to differentiate the product of two functions.
The product rule: \(y = uv\),
\(\dfrac{dy}{dx} = u\dfrac{dv}{dx} + v\dfrac{du}{dx} \)
Or \([f(x)g(x)]' = f(x)g'(x) + f'(x)g(x) \)
#9.13
Be able to differentiate the quotient of two functions.
The quotient rule: \(y = \dfrac{u}{dv}\),
\(\dfrac{dy}{dx} = \dfrac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}} {v^2} \)
\(\Bigg[\dfrac{f(x)}{g(x)}\Bigg]' = \dfrac{f'(x)g(x) - f(x)g'(x)}{\big(g(x)\big)^2} \)
#9.14
Be able to differentiate composite functions using the chain rule.
\(y=f(u)\), \(u=g(x)\),
\(\dfrac{dy}{dx} = \dfrac{dy}{du} × \dfrac{du}{dx} \)
Or \(\big\{f\big[g(x)\big]\big\}' = f'\big[g(x)\big]g'(x) \)
#9.15
Be able to find rates of change using the chain rule, including connected rates of change and differentiation of inverse functions.
\(\dfrac{dy}{dx} = \dfrac{1}{\Big(\dfrac{dy}{dx}\Big)} \)
#9.16
Be able to differentiate a function or relation defined implicitly.
e.g. \((x + y)^2 = 2x\).
[Excludes: Second and higher derivatives.]
#9.17
Understand that a section of curve which has increasing gradient (and so positive second derivative) is concave upwards.
Understand that a section of curve which has decreasing gradient (and so negative second derivative) is concave downwards.
\(\huge\smallsmile\) concave upwards (convex downwards)
\(\huge\smallfrown\) concave downwards (convex upwards)
Notation: The wording “concave upwards” or “concave downwards” will be used in examination questions.
#9.18
Understand that a point of inflection on a curve is where the curve changes from concave upwards to concave downwards (or vice versa) and hence that the second derivative at a point of inflection is zero.
Be able to use differentiation to find stationary and non-stationary points of inflection.
Learners are expected to be able to find and classify points of inflection as stationary or non-stationary.
Distinguish between maxima, minima and stationary points of inflection.
#9.19
Know that integration is the reverse of differentiation.
Fundamental Theorem of Calculus.
#9.20
Be able to integrate functions of the form \(kx^n\) where \(k\) is a constant and \(n \neq -1\).
Including related sums and differences.
#9.21
Be able to find a constant of integration given relevant information.
e.g. Find \(y\) as a function of \(x\) given that \(\dfrac{dy}{dx} = x^2 + 2\)and \(y = 7\) when \(x = 1\).
#9.22
Know what is meant by indefinite and definite integrals.
Be able to evaluate definite integrals.
e.g. \(\displaystyle\int_1^3{(3x^2 + 5x - 1)}~dx\).
#9.23
Be able to use integration to find the area between a graph and the x-axis.
Includes areas of regions partly above and partly below the x-axis.
General understanding that the area under a graph can be found as the limit of a sum of areas of rectangles.
[Excludes: Formal understanding of the continuity conditions required for the Fundamental Theorem of Calculus.]
#9.24
Be able to integrate \(e^{kx}\), \(\dfrac{1}{x}\), \(\sin{kx}\), \(\cos{kx}\) and related sums, differences and constant multiples.
\(\displaystyle\int{\dfrac{1}{x}}~dx = \ln{|x|} + c, x \neq 0 \)
\(x\) in radians for trigonometrical integrals.
[Excludes: Integrals involving inverse trigonometrical functions.]
#9.25
Understand integration as the limit of a sum.
Know that \(\lim\limits_{\delta x→0} \displaystyle\sum_a^b f(x) \delta x = \displaystyle\int_a^b{f(x)} dx\)
#9.26
Be able to use integration to find the area between two curves.
Learners should also be able to find the area between a curve and the \(y\)-axis, including integrating with respect to \(y\).
#9.27
Be able to use integration by substitution in cases where the process is the reverse of the chain rule (including finding a suitable substitution).
e.g. \((1+2x)^8\), \(x(1+x^2)^8\), \(xe^{x^2}\), \(\dfrac{1}{2x+3}\)
Learners can recognise the integral, they need not show all the working for the substitution.
#9.28
Be able to use integration by substitution in other cases.
Learners will be expected to find a suitable substitution in simple cases e.g. \(\dfrac{x}{(x+1)^3}\).
[Excludes: Integrals requiring more than one substitution before they can be integrated.]
#9.29
Be able to use the method of integration by parts in simple cases.
Includes cases where the process is the reverse of the product rule, e.g. \(xe^x\). More than one application of the method may be required.
Includes being able to apply integration by parts to \(\ln{x}\).
[Excludes: Reduction formulae.]
#9.30
Be able to integrate using partial fractions that are linear in the denominator.
#9.31
Be able to formulate first order differential equations using information about rates of change.
Contexts may include kinematics, population growth and modelling the relationship between price and demand.
#9.32
Be able to find general or particular solutions of first order differential equations analytically by separating variables.
Equations may need to be factorised using a common factor before variables can be separated.
#9.33
Be able to interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution.
Includes links to kinematics.