What does the domain represent in mathematical modelling ?

In mathematical modelling , the domain represents the reasonable range of input values , considering the real-life context of the situation being modelled.

How can unknown parameters be found in mathematical models ?

Unknown parameters in mathematical models can be found by forming equations , substituting given values and solving the equations simultaneously. You will frequently use a graphing calculator (GDC) to help with this.

What is extrapolation in mathematical modelling ?

Extrapolation is making predictions outside the range of the data.

True or False? Extrapolation is an accurate method for predicting new values in mathematical modelling.

False. Extrapolation is not considered to be an accurate method for predicting new values in mathematical modelling.

What is a key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation ?

A key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation is to restrict the domain appropriately.

True or False? Exponential models are commonly used for arithmetic sequences .

False. Exponential models are not used for arithmetic sequences . Linear models are used for arithmetic sequences, while exponential models are used for geometric sequences.

What type of model is typically used for projectile motion ?

A quadratic model is typically used to model projectile motion.

How can you estimate outputs using a model?

You can estimate outputs by substituting values for the input variable into the model.

What does substituting t = 0 into a time-based model typically give you?

Substituting t = 0 into a time-based model typically gives you the initial value according to the model.

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FrontExponential & Logarithmic Functions
What is the range of an exponential function $f (x) = a^{x}$ , where $a > 0$ ?

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What is the range of an exponential function $f (x) = a^{x}$ , where $a > 0$ ?
The range of an exponential function $f (x) = a^{x}$ , where $a > 0$ , is $f (x) > 0$ .
True or False?
The domain of $f (x) = a^{x}$ , where $a > 0$ , is $x \in ℝ$ .
True.
The domain of $f (x) = a^{x}$ , where $a > 0$ , is $x \in ℝ$ .
True or False?
$a^{x} = e^{a \ln x}$ .
False.
$a^{x} = e^{x \ln a}$
This is given in the exam formula booklet.
What does the graph of $f (x) = a^{x}$ look like when $a > 1$ ?
$f (x) = a^{x}$ is an increasing function when $a > 1$ .
The equation of the graph below is $y = a^{x}$ . What are the possible values for $a$ ?
The exponential graph is decreasing, therefore $0 < a < 1$ .
Every exponential graph of the form $f (x) = a^{x}$ where $a > 0$ passes through which point?
Every exponential graph of the form $f (x) = a^{x}$ where $a > 0$ passes through the point (0, 1).
What is the equation of the asymptote for every exponential graph of the form $f (x) = a^{x}$ where $a > 0$ ?
The equation of the asymptote for every exponential graph of the form $f (x) = a^{x}$ where $a > 0$ is $y = 0$ .
What is the inverse function of $f (x) = a^{x}$ where $a > 0$ ?
The inverse function of $f (x) = a^{x}$ where $a > 0$ is $f^{- 1} (x) = \log_{a} x$ .
What is the domain of the logarithmic function $f (x) = \log_{a} x$ where $a > 0$ ?
The domain of the logarithmic function $f (x) = \log_{a} x$ where $a > 0$ is $x > 0$ .
True or False?
The graph of $y = \log_{a} x$ where $a > 0$ crosses the coordinate axes at the point (0, 1).
False.
The graph of $y = \log_{a} x$ where $a > 0$ crosses the the coordinate axes at the point (1, 0) not (0, 1).
What does $\log_{a} (a^{x})$ simplify to when $a > 0$ ?
$\log_{a} (a^{x})$ simplifies to $x$ when $a > 0$ .
This is because the logarithmic function and exponential function are inverses.
What is the equation of the asymptote for every logarithmic graph of the form $f (x) = \log_{a} x$ where $a > 0$ ?
The equation of the asymptote for every logarithmic graph of the form $f (x) = \log_{a} x$ where $a > 0$ is $x = 0$ .
What does the graph of $f (x) = \log_{a} x$ look like when $a > 1$ ?
The graph of $f (x) = \log_{a} x$ is increasing when $a > 1$ .
True or False?
$\sqrt{x} = x - 2$ has the same solutions as $x = {(x - 2)}^{2}$ .
False.
$\sqrt{x} = x - 2$ does not have the same solutions as $x = {(x - 2)}^{2}$ .
When you square both sides of an equation, you can introduce additional solutions.
$\sqrt{x} = x - 2$ has one solution, $x = 4$ , whereas $x = {(x - 2)}^{2}$ has two solutions, $x = 1$ or $x = 4$ .
What substitution could you use to turn $2 x^{6} + 3 x^{3} - 4 = 0$ into a quadratic equation?
You could use the substitution $y = x^{3}$ to turn $2 x^{6} + 3 x^{3} - 4 = 0$ into the quadratic equation $2 y^{2} + 3 y - 4 = 0$ .
True or False?
The first step to solving the equation $e^{x} + e^{- x} = 2$ , without a GDC, is to take natural logarithms of each term, $\ln (e^{x}) + \ln (e^{- x}) = \ln (2)$ .
False.
You can not take natural logarithms of each term of an equation, you can only take natural logarithms of both sides of the equation.
The first step to solving the equation $e^{x} + e^{- x} = 2$ is to transform it into a quadratic equation using the substitution $y = e^{x}$ .
How could you use a graph to solve the equation $f (x) = g (x)$ ?
To solve the equation $f (x) = g (x)$ , you could:
- sketch the graphs of $y = f (x)$ and $y = g (x)$ on your GDC,
- find the points of intersection,
- the $x$ -coordinates are the solutions to the equation.
How could you use the graph of $y = f (x) - g (x)$ to find the solutions to the equation $f (x) = g (x)$ ?
To find the solutions to the equation $f (x) = g (x)$ , you could find the x-intercepts (roots) of the graph of $y = f (x) - g (x)$ .
True or False?
The solutions to the equation $(x - 1) (x^{2} + 2) = (x - 1) (3 x + 6)$ are the same as the solutions to the equation $x^{2} + 2 = 3 x + 6$ .
False.
The solutions to the equation $(x - 1) (x^{2} + 2) = (x - 1) (3 x + 6)$ are not the same as the solutions to the equation $x^{2} + 2 = 3 x + 6$ .
The first equation also has a solution of $x = 1$ .
Dividing both sides of an equation by something that could equal zero can cause you to lose solutions.
What would be the first step to solving an equation such as $\log_{2} (x - 1) = 2 - \log_{2} (x + 2)$ without a GDC?
The first step to solving an equation such as $\log_{2} (x - 1) = 2 - \log_{2} (x + 2)$ would be to rearrange to get all the logarithms on one side of the equation such as $\log_{2} (x - 1) + \log_{2} (x + 2) = 2$ .
Once the equation is in that form, laws of logarithms can be used.
What does the domain represent in mathematical modelling?
In mathematical modelling, the domain represents the reasonable range of input values, considering the real-life context of the situation being modelled.
How can unknown parameters be found in mathematical models?
Unknown parameters in mathematical models can be found by forming equations, substituting given values and solving the equations simultaneously.
You will frequently use a graphing calculator (GDC) to help with this.
What is extrapolation in mathematical modelling?
Extrapolation is making predictions outside the range of the data.
True or False?
Extrapolation is an accurate method for predicting new values in mathematical modelling.
False.
Extrapolation is not considered to be an accurate method for predicting new values in mathematical modelling.
What is a key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation?
A key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation is to restrict the domain appropriately.
True or False?
Exponential models are commonly used for arithmetic sequences.
False.
Exponential models are not used for arithmetic sequences.
Linear models are used for arithmetic sequences, while exponential models are used for geometric sequences.
What type of model is typically used for projectile motion?
A quadratic model is typically used to model projectile motion.
How can you estimate outputs using a model?
You can estimate outputs by substituting values for the input variable into the model.
What does substituting t = 0 into a time-based model typically give you?
Substituting t = 0 into a time-based model typically gives you the initial value according to the model.

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Other Functions & Graphs (DP IB Analysis & Approaches (AA))

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Cards in this collection (29)

1. Number & Algebra

Number & Algebra Toolkit

Exponentials & Logs

Sequences & Series

Simple Proof & Reasoning

Proof by Induction & Contradiction

Binomial Theorem

Permutations & Combinations

Complex Numbers

Further Complex Numbers

Systems of Linear Equations

2. Functions

Linear Functions & Graphs

Quadratic Functions & Graphs

Functions Toolkit

Other Functions & Graphs

Reciprocal & Rational Functions

Transformations of Graphs

Polynomial Functions

Inequalities

Modulus Functions & Further Transformations

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

The Unit Circle & Exact Values

Trigonometric Functions & Graphs

Trigonometric Equations & Identities

Inverse & Reciprocal Trigonometric Functions

Trigonometric Proof & Equation Strategies

Vector Properties

Vector Equations of Lines

Vector Planes

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Probability

Discrete Random Variables

Binomial Distribution

Normal Distribution

Continuous Random Variables

5. Calculus

Differentiation

Techniques & Applications of Differentiation

Integration

Techniques & Applications of Integration

Optimisation

Kinematics

Basic Limits & Continuity

Further Differentiation

Further Integration

Differential Equations

Maclaurin Series

Limits using l'Hôpital's Rule & Maclaurin Series