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.
![](https://kognity-prod.imgix.net/media/edusys_2/content_uploads/Core5_1_2step2.e5dc655f6fb83d910baa.png)
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.
![](https://kognity-prod.imgix.net/media/edusys_2/content_uploads/ibdp.maths.core.book.05.01.aw.18.0f01fef4f5bfdb4c84b6.png)
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
![](https://kognity-prod.imgix.net/media/edusys_2/content_uploads/Core5_1_4step3.044d256d2737760f6489.png)
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
Applications
Kinematics
Displacement (s), Velocity (v), Acceleration (a)
![](http://ibmathstuff.wdfiles.com/local--files/kinematics/KinematicRelationships.jpg)
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-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: .
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 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
![](https://kognity-prod.imgix.net/media/edusys_2/content_uploads/Core5_5_1step7.f9e02d79595009dbce2f.png)
![](https://kognity-prod.imgix.net/media/edusys_2/content_uploads/Core5_5_1step2.1a79b06d4621a115341b.png)
Numerical Solve
![](https://kognity-prod.imgix.net/media/edusys_2/content_uploads/Core5_5_2step1.7b5a72d4426ee5278312.png)
![](https://kognity-prod.imgix.net/media/edusys_2/content_uploads/Core5_5_2step2.7a52961ff0fda31c3d5f.png)
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 , then: $$y' = uv' + u'v$$
Chain Rule
Chain Rule where , then: $$\frac{dy}{dx} = \frac{dy}{du} \times \frac {du}{dx}$$
![](https://ibrecap.com/images/user_images/Annotation 2020-03-04 1100581583316148.png)
Turning points
Point |
|
|
|
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:
- y-intercept:
Turning points:
- Minima: and
- Maxima: and
- Point of inflection:
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.
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