# Subject content

This is an extract of the full specification, which you can download from this page.

## 3.7 FSMQ Calculus (9998)

### Calculus (9998)

This qualification has been developed to allow you to demonstrate your ability to use

• differentiation
• integration
• differential equations

to analyse, make sense of and describe real world situations and to solve problems. You will also investigate the use of numerical methods to find gradients and evaluate integrals and compare these with analytic methods.

 Before you start this qualification You must: This includes: be able to use algebraic methods to rearrange and solve linear and quadratic equations Solution of a quadratic equation by at least one of the following methods: use of a graphics calculatoruse of formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$(which must be memorised)completing the squareSolution by factorisation will be acceptable where the quadratic factorises. have knowledge of basic functions and how geometric transformations can be applied to them using transformations by the vector $\left [ \begin{matrix} a \\ 0 \end{matrix} \right ] \space$ and by the vector $\space \left [ \begin{matrix} 0 \\ a \end{matrix} \right ]$stretches of scale factor $a$ with the invariant line $\space x = 0 \space$ and with the invariant line $\space y = 0$ being familiar with graphs and functions of: powers of $\space x$ , eg $\space y = kx^{-2}; \space y = kx^{-1};$$\space y = kx^{\frac{1}{2}}; \space y = kx^{3}; \space y = kx^4$quadratics: $y = ax^2 + bx + c$$y = (ax - b)(x - c)$trigonometric functions: $y = A \sin (mx + c)$$y = A \cos (mx + c)$exponential functions: $y = k\mbox {e}^{mx}$( $m \space$ positive or negative)logarithmic functions: $y = a \ln (bx)$

### Using calculators and computers

Using calculators and computers

When carrying out calculations, you may find the use of a standard scientific calculator sufficient.

You should learn to use your calculator effectively and efficiently. This will include learning to use:

• memory facilities
• function facilities (e.g., $\mbox {e}^x, \space \sin(x), \space ...$ )

It is important that you are also able to carry out certain calculations without using a calculator, using both written methods and 'mental' techniques.

Whenever you use a calculator you should record your working as well as the result.

### Understanding and using differentiation

 Understandingandusingdifferentiation You should learn to: This includes: understand and calculate gradient at a point, $\space a \space$ , on a function $\space y = \mbox {f}(x) \space$ using the numerical approximation: $\text{gradient} \approx \frac{\mbox {f}(a + h) - \mbox {f}(a)}{h}$where $\space h$ is small understanding how to improve the calculation of gradient at a point by using a smaller interval, $\space h.$ understand and interpret gradients in terms of their physical significance understand and use the correct units with which to measure gradients /rates of change sketch graphs of gradient functions curves that you do not know as functionscurves defined as functions identify the key features of gradient functions in terms of the gradient of the original function zeros of gradient functions linking to local turning points understand how $\space \frac{\mbox {f}(x+h) - \mbox {f}(x)}{h} \space$ can be used to generate a gradient function differentiate functions using notations $\space \frac{\mbox {d}y}{\mbox {d}x} \space$ and $\space \mbox {f}'(x) \space$polynomialstrigonometric functions using radiansexponential functions Differentiate sums and differences of functionsfunctions multiplied by a constantproducts of functions find the second derivatives of functions using notations $\space \frac{\mbox {d}^2y}{\mbox {d}x^2} \space$ and $\space \mbox {f}''(x)$ identify the key features of a second derivative linking positive values to increasing gradientlinking negative values to decreasing gradientlinking zero values to points of inflexion Applications of differentiation to gradients, maxima and minima and stationary points, increasing and decreasing functions Application to determining maxima and minimaunderstanding the importance of the second derivative and its value at such pointsunderstanding that zero values of the second derivative can occur at maximum and minimum points as well as points of inflexion

### Understanding and using integration

 Understandingandusingintegration You should learn to: This includes: estimate areas under graphs of functions using numerical methods the trapezium ruleunderstanding how to improve your calculation of the area under a graph by using a smaller interval. understand and find areas under curves, between $\space x = a \space$ and $\space x = b \space$ using $\space \int_a^b \mbox {f}(x)\mbox {d}x \space$ , $\space (\mbox {f}(x) \geq 0) \space$ understand integration as the reverse process of differentiation understand and determine indefinite integrals of functions $x^n \space$ (including $\space n = -1 \space$ and fractional)$A \sin (mx+c) \space$$A \cos (mx+c) \space$$k\mbox {e}^{mx} \space$ ( $m$ positive or negative)sums, differences and constant multiples of these using a constant of integration Integration by inspection and by one use of integration by parts eg: $\int e^{-5x}dx$, $\int \mbox {sin}6xdx$  eg: $\int xe^{-5x}dx$,$\int x\mbox {cos}4xdx$, $\int x\mbox {ln}xdx$ understand the idea of constant of integration and be able to calculate this in known situations be able to determine definite integrals for functions those functions defined above

### Understanding and using differential equations

 Understandingandusingdifferentialequations You should learn to: This includes: find families of solutions to first order differential equations with separable variables find particular solutions when boundary conditions are given