True or False? A logarithmic graph does not have a y -intercept .

True. A logarithmic graph does not have a y -intercept . There is a vertical asymptote at the y -axis.

True or False? A logistic graph has no roots .

True. A logistic graph has no roots .

In what situation can a natural logarithmic model be used?

A natural logarithmic model can be used when: the variable increases rapidly for a period, followed by a much slower rate of increase with no limiting value . E.g. sound intensity level can be modelled by a natural logarithmic function.

What is a possible limitation of a natural logarithmic model?

A possible limitation of a natural logarithmic model is that it is unbounded . In real-life, the variable may have a limiting value.

Give an example of a situation that could be modelled by a piecewise linear function.

A piecewise linear function can be used to model a situation where there is a constant rate of change that is different for different intervals . For example, a taxi charge that doubles after midnight, or a car rental fee that triples during national holidays.

What is a limitation of piecewise linear models ?

One limitation of piecewise linear models is that they assume a constant rate of change in each interval . In reality , there may be a continuously variable rate of change or gradual transitions between rates.

True or False? Individual functions within a piecewise function must always be linear .

False. Individual functions within a piecewise function do not have to be linear, they can contain any type of function , e.g. exponential, quadratic, ...

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IBMathsDPApplications & Interpretation (AI)HLFlashcards2. FunctionsModelling with Logarithmic, Logistic & Piecewise Functions

Modelling with Logarithmic, Logistic & Piecewise Functions (DP IB Applications & Interpretation (AI))

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What is the general form for a logarithmic function?

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What is the general form for a logarithmic function?
The general form for a logarithmic function is $f (x) = a + b \ln x$ , $x > 0$ .
True or False?
$\ln x \equiv \log_{e} (x)$
True.
$\ln x \equiv \log_{e} (x)$
This is the inverse of $f (x) = e^{x}$ .
Remember that $\ln (e^{x}) = x$ and $e^{\ln x} = x$ .
Which point will a logarithmic graph always pass through?
A logarithmic graph always pass through the point $(1, a)$ .
True or False?
A logarithmic graph does not have a y-intercept.
True.
A logarithmic graph does not have a y-intercept.
There is a vertical asymptote at the y-axis.
What are the coordinates of the single root of a logarithmic graph, $f (x) = a + b \ln x$ , $x > 0$ ?
The coordinates of the single root of a logarithmic graph, $f (x) = a + b \ln x$ , $x > 0$ are $(e^{- \frac{a}{b}}, 0)$ .
This is found by solving $a + b \ln x = 0$
For a logarithmic graph, $f (x) = a + b \ln x$ , $x > 0$ , what does a positive value of $b$ indicate?
For a logarithmic graph, $f (x) = a + b \ln x$ , $x > 0$ , a positive value of $b$ indicates that the graph is increasing.
What is the general form for a logistic function?
The general form for a logistic function is $f (x) = \frac{L}{1 + C e^{- k x}}$ , where $L, C$ and $k$ are positive constants.
True or False?
A logistic graph shows both increases and decreases.
False.
A logistic graph is always increasing.
What is the range of a logistic function, $f (x) = \frac{L}{1 + C e^{- k x}}$ ?
The range of a logistic function, $f (x) = \frac{L}{1 + C e^{- k x}}$ , is any real positive values less than $L$ .
What are the coordinates of the y-intercept of a logistic graph, $f (x) = \frac{L}{1 + C e^{- k x}}$ ?
The coordinates of the y-intercept of a logistic graph, $f (x) = \frac{L}{1 + C e^{- k x}}$ , are $(0, \frac{L}{1 + C})$ .
True or False?
A logistic graph has no roots.
True.
A logistic graph has no roots.
What are the equations of the two horizontal asymptotes of a logistic graph, $f (x) = \frac{L}{1 + C e^{- k x}}$ ?
The equations of the two horizontal asymptotes of a logistic graph, $f (x) = \frac{L}{1 + C e^{- k x}}$ , are:
- $y = L$ ,
- and $y = 0$ .
In what situation can a natural logarithmic model be used?
A natural logarithmic model can be used when:
- the variable increases rapidly for a period,
- followed by a much slower rate of increase with no limiting value.
E.g. sound intensity level can be modelled by a natural logarithmic function.
In a natural logarithmic model, $f (x) = a + b \ln x$ , what does the value $a$ represent?
In a natural logarithmic model, $f (x) = a + b \ln x$ , the value $a$ represents the value of the function when $x = 1$ .
In a natural logarithmic model, $f (x) = a + b \ln x$ , what does the value $b$ represent?
In a natural logarithmic model, $f (x) = a + b \ln x$ , the value $b$ represents the rate of change of the function.
What is a possible limitation of a natural logarithmic model?
A possible limitation of a natural logarithmic model is that it is unbounded.
In real-life, the variable may have a limiting value.
In what situation can a logistic model be used?
A logistic model can be used when:
- the variable initially increases exponentially,
- and then tends towards a limit.
E.g. the number of bacteria on an apple $t$ seconds after removing it from protective packaging.
In a logistic model, $f (x) = \frac{L}{1 + C e^{- k x}}$ , what does the value $L$ represent?
In a logistic model, $f (x) = \frac{L}{1 + C e^{- k x}}$ , the value $L$ represents the limiting capacity.
This is the value that the model tends to as x gets large.
In a logistic model, $f (x) = \frac{L}{1 + C e^{- k x}}$ , what does the value $k$ represent?
In a logistic model, $f (x) = \frac{L}{1 + C e^{- k x}}$ , the value $k$ determines the rate of increase of the model.
In a logistic model, $f (x) = \frac{L}{1 + C e^{- k x}}$ , what determines the initial value?
In a logistic model, $f (x) = \frac{L}{1 + C e^{- k x}}$ , the value $C$ (along with $L$ ), is used to determine the initial value of the model.
The initial value is given by $\frac{L}{1 + C}$ .
What are the two possible limitations of a logistic model?
The two possible limitations of a logistic model are:
1. A logistic graph is bounded by the limit $L$ , however, in real-life the variable might be unbounded.
2. A logistic graph is always increasing, however, in real-life there could be periods where the variable decreases or fluctuates.
Give an example of a situation that could be modelled by a piecewise linear function.
A piecewise linear function can be used to model a situation where there is a constant rate of change that is different for different intervals.
For example, a taxi charge that doubles after midnight, or a car rental fee that triples during national holidays.
What is a limitation of piecewise linear models?
One limitation of piecewise linear models is that they assume a constant rate of change in each interval.
In reality, there may be a continuously variable rate of change or gradual transitions between rates.
True or False?
Individual functions within a piecewise function must always be linear.
False.
Individual functions within a piecewise function do not have to be linear, they can contain any type of function, e.g. exponential, quadratic, ...
Why might a non-linear piecewise function be used to model the height $H (t)$ of water in a bathtub with after $t$ minutes?
A non-linear piecewise function be used to model the height $H (t)$ of water in a bathtub with after $t$ minutes as the rate of change of the height of the water over time may differ depending on the shape of the bath tub.
E.g. if the bath tub has curved sides at the base, then a cubic function may be suitable to model the height of the water, but when the sides go straight, a linear model may be more suitable.
True or False?
When graphed, the individual functions within a piecewise function should join to make a continuous graph.
True.
When graphed, the individual functions within a piecewise function should join to make a continuous graph.
E.g. if $f (x) = \{\begin{matrix} f_{1} (x) \\ f_{2} (x) \end{matrix} \begin{matrix} a \leq x < b \\ b \leq x < c \end{matrix}$ then $f_{1} (b) = f_{2} (b)$

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Modelling with Logarithmic, Logistic & Piecewise Functions (DP IB Applications & Interpretation (AI))

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1. Number & Algebra

Number Toolkit

Exponentials & Logs

Sequences & Series

Financial Applications

Complex Numbers

Further Complex Numbers

Matrices

Eigenvalues & Eigenvectors

2. Functions

Linear Functions & Graphs

Further Functions & Graphs

Modelling with Functions

Functions Toolkit

Transformations of Graphs

Modelling with Logarithmic, Logistic & Piecewise Functions

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

Trigonometric Identities & Equations

Voronoi Diagrams

Matrix Transformations

Vector Properties

Vector Equations of Lines

Modelling with Vectors

Graph Theory

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Non-linear Regression

Probability

Probability Distributions

Random Variables

Binomial Distribution

Normal Distribution

Combinations of Normal Distributions & Sample Mean Distributions

Poisson Distribution

Hypothesis Testing using the Chi-squared Distribution

Hypothesis Testing for Population Parameters

Transition Matrices & Markov Chains

5. Calculus

Differentiation

Further Differentiation

Integration

Further Integration

Kinematics

Differential Equations

Coupled & Second Order Differential Equations