# Concept of a limit

The number \(A\) is called the** limit** of the sequence \(\{a_n\}\) if \(a_n\) is close to \(A\) for large values of n.

The number \(A\) is called the limit (at infinity) of the function f if \(f(x)\) is close to \(A\) for large values of x.

Note that this is closely related to the concept of a horizontal asymptote. If \(A\) is the limit (at infinity) of \(f\), then the line \(y=A\) is a horizontal asymptote of the graph of \(y=f(x)\).

Note that the function does not have to be defined at x=ax=a in order to have a limit. If the function happens to be defined at \(x=a\) and the limit is the same as \(f(a)\) we say that the function is **continuous **at \(x=a\).

# The gradient at a point

**Differentiation** is a way to find the gradient (change in x over the change in y) of a function at any point, written as:

$$ f'(x) \space, \space y' \space, \space \frac{dy}{dx} $$

**Tangent line to a point on a curve** is a linear line with the same gradient as that point on the curve.

Use your calculator to find the gradient of the graph of \(y=ln(3x2+2)\) at \(x=1\).

# Gradient function

Common forms of notation for the derivative are \(f'\), \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) and \(\dfrac{\mathrm{d}}{\mathrm{d}x}f\).

The diagram below shows the graph of the function f.

Given that \(f'(a)=−1\), \(f'(b)=0\) and \(f'(c)=3\).

Find the value of a, of b, and of c.

#### Ans: \(a=1\), \(b=2\), \(c=−1\)

## Graphing the Derivative

The function \(F\) is an anti-derivative of the function f if \(F'(x)=f(x)\).

The collection of all anti-derivatives of the function f is denoted by \( \displaystyle{\int f(x)\mathrm{d}x}\).

This is also called the indefinite integral of \(f\).

### \(\displaystyle{ \int ax^n\mathrm{d}x=\dfrac{a}{n+1}x^{n+1}+c, }\) where c is any constant and \(n\ne -1\) is an integer.

^Given in the formula booklet

# Integral of the sum or difference

### \(\displaystyle{ \int f(x)+g(x)\mathrm{d}x=\int f(x)\mathrm{d}x+\int g(x)\mathrm{d}x }\)

# Applications

## Kinematics

Displacement (s), Velocity (v), Acceleration (a)

# Graph properties

By the end of this subtopic you should be able to:

- identify properties of graphs based on the information given by the derivative:
- identify intervals where the function is increasing
- identify intervals where the function is decreasing
- identify the x$x$-coordinates of turning points.

# The power rule

### If \(f(x)=x^n\), then \(f′(x)=nx^{n−1}\)

This rule is in the formula booklet.

# The constant factor rule

### If \(f(x)=ax^n\), then \(f'(x)=nax^{n−1}\).

This is not in the formula booklet. You need to remember this rule and, more importantly, make sure that you know how to use it.

# The sum rule

### If \(h(x)=f(x)+g(x)\), then \(h'(x)=f'(x)+g'(x)\).

This is not in the formula booklet. You need to remember this rule and, more importantly, make sure that you know how to use it.

# The normal

### \(M_Normal = - \frac{1}{M_Tangent}

The line that is perpendicular to the tangent to a curve at the point of tangency is called the **normal**.

**Tangent** - line with the same gradient as a point on a curve.

**Normal** - perpendicular to the tangent at a point

Both are linear lines with the general formula: $y=mx+c$.

## Finding the Normal

- Use the derivative to find the gradient of the tangent. For normal

then do $$ m = \frac{−1}{slope \space of \space tangent}$$ - Input the x-value of the point into f (x) to find y.
- Input y, m and the x-value into $y=mx+c$ to find c.

# The area under a curve

The formula booklet gives the following:

The area between a curve \(y=f(x)\) and the x-axis, where \(f(x)>0\) is

\(A = \displaystyle{\int_a^b y\,\mathrm{d}x.}\)

## Graphical Solve

## Numerical Solve

# Further differentiation

## Exponential Function

If \(f(x)=k\mathrm{e}^{ax+b}\), then the derivative is \(f'(x)=ka\mathrm{e}^{ax+b}\).

## Product Rule

**Product Rule** $y=uv$, then: $$y' = uv' + u'v$$

## Chain Rule

**Chain Rule** $y=g\left(u\right)$ where $u=f\left(x\right)$, then: $$\frac{dy}{dx} = \frac{dy}{du} \times \frac {du}{dx}$$

# Turning points

Point | $f\left(x\right)=$ | $f\text{'}\left(x\right)=$ | $f\text{'}\text{'}\left(x\right)$ |

Local Min | 0 | Positive Number | |

Local Max | 0 | Negative Number | |

Point of Inflection | 0 | ||

Colour | Green | Orange | Red |

\graph{{-0.1x^{3}+3x:green},{-0.3x^{2}+3:orange},{-0.6x:red},{x=-3.162:black:dashed},{x=3.162:black:dashed}}

# Sketching Graphs

Gather information before sketching:

## Intercepts:

- x-intercept: $f\left(x\right)=0$
- y-intercept: $f\left(0\right)$

## Turning points:

- Minima: $f\text{'}\left(x\right)=0$ and $f\text{'}\text{'}\left(x\right)<0$
- Maxima: $f\text{'}\left(x\right)=0$ and $f\text{'}\text{'}\left(x\right)>0$
- Point of inflection: $f\text{'}\text{'}\left(x\right)=0$

## Asymptotes

- Vertical: x-value when the function divides by 0
- Horizontal: y-value when x → ∞

Plug the found x-values into f (x) to determine the y-values.

# Editors

- CD_FER - 979 words.
- joeClinton - 28 words.

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