A limit is a value that a function approaches as x approaches a particular value from either side.

What is a derivative?

A derivative (also known as a gradient function ) is a function that relates the gradient of another function to the value of x .

True or False? The derivative of a constant function is always zero .

True. The derivative of a constant function is always zero .

A tangent is a straight line that touches a curve at a point without crossing through it.

True or False? The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.

True. The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.

What is the derivative of a sum (or difference ) of functions equal to?

The derivative of a sum (or difference ) of functions is equal to the sum (or difference) of their individual derivatives.

What is a normal (or normal line ) with regard to a function?

A normal (or normal line ) is a straight line that passes through a point on a curve and is perpendicular to the tangent at that point.

What is the relationship between the gradients of a tangent and a normal at the same point on a curve?

At the same point on a curve, the product of the gradients of a tangent and its normal is equal to -1.

True or False? Local minimum and maximum points are types of stationary points .

True. Local minimum and maximum points are types of stationary points .

What is a local minimum point ?

A local minimum point is a point where the function value is the lowest in the immediate vicinity .

True or False. A local maximum point is the point at which a function takes on its maximum value.

False. A local maximum point is not necessarily the point at which a function takes on its maximum value . A local maximum point is a point at which the function value is the highest in the immediate vicinity . There may however be other points (not in the immediate vicinity) at which the function takes on a higher value.

How can you use gradients to determine if a stationary point is a local minimum or local maximum ?

To determine if a stationary point is a local minimum or local maximum , look at the gradient of the function on either side of the stationary point. If the gradient is positive to the left of the point and negative to the right , then the stationary point is a local maximum . If the gradient is negative to the left of the point and positive to the right , then the stationary point is a local minimum .

What are optimisation problems ?

Optimisation problems are problems that involve maximizing or minimising a quantity.

What is a constraint?

A constraint is a condition that limits the possible values of variables in a problem.

True or False? A useful first step in solving an optimisation problem is rewriting the quantity to be optimised as a function of a single variable .

True. A useful first step in solving an optimisation problem is rewriting the quantity to be optimised as a function of a single variable .

True or False? A GDC can be used to find local maximum or minimum points in optimisation problems.

True. A GDC can be used to find local maximum or minimum points in optimisation problems. One way to do this is to use the GDC's graphing features to draw a graph of the function.

True or False? The context of a question is not important when writing down the final solution of an optimisation problem.

False. The final step in solving an optimisation problem is interpreting the answer in the context of the question.

True or False? Optimisation problems always have a unique solution.

False. Optimisation problems may have multiple solutions or no solution .

Why is it important to interpret the solution to an optimisation problem in the context of the problem?

Interpreting the solution to an optimisation problem in the context of the problem is important to ensure the answer makes sense in the real-world situation being modelled .

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What is a limit?

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What is a limit?
A limit is a value that a function approaches as x approaches a particular value from either side.
What is a derivative?
A derivative (also known as a gradient function) is a function that relates the gradient of another function to the value of x.
What does $\frac{d y}{d x}$ mean?
$\frac{d y}{d x}$ means the derivative of y with respect to x.
What does $f^{'} (x)$ mean?
$f^{'} (x)$ means the derivative of the function $f (x)$ with respect to $x$ .
True or False?
The derivative of a constant function is always zero.
True.
The derivative of a constant function is always zero.
State the formula for differentiating $x^{n}$ .
If $f (x) = x^{n}$ , then $f^{'} (x) = n x^{n - 1}$ .
I.e. the original power comes in front as a multiplier, then the original power is reduced by 1.
This formula is in the exam formula booklet.
What is the derivative of $a x$ , where $a$ is a constant?
The derivative of $a x$ , where $a$ is a constant, is $a$ .
What is a tangent?
A tangent is a straight line that touches a curve at a point without crossing through it.
True or False?
The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.
True.
The gradient of a curve at a point is equal to the gradient of the tangent to the curve at that point.
What is an alternative notation for $f^{'} (x)$ ?
If $y = f (x)$ , then an alternative notation for $f^{'} (x)$ is $\frac{d y}{d x}$ .
True or False?
The derivative of $\frac{4}{x}$ is $4 x^{- 2}$ .
False.
The derivative of $\frac{4}{x}$ is $- 4 x^{- 2}$ , which is the same as $- \frac{4}{x^{2}}$ .
To differentiate a reciprocal, first rewrite it as a negative power: $\frac{4}{x} = 4 x^{- 1}$ .
Then use "If $f (x) = x^{n}$ , then $f^{'} (x) = n x^{n - 1}$ ".
What is the derivative of a sum (or difference) of functions equal to?
The derivative of a sum (or difference) of functions is equal to the sum (or difference) of their individual derivatives.
True or False?
The formula for differentiating powers of $x$ , that says if $f (x) = x^{n}$ then $f^{'} (x) = n x^{n - 1}$ , can only be used when the power $n$ is an integer.
False.
The formula for differentiating powers of $x$ , that says if $f (x) = x^{n}$ then $f^{'} (x) = n x^{n - 1}$ , can be used when the power $n$ is any rational number.
What does it mean if $f^{'} (x) > 0$ ?
If $f^{'} (x) > 0$ , it means the function $f (x)$ is increasing.
True or False?
A function is decreasing if $f^{'} (x) < 0$ .
True.
A function is decreasing if $f^{'} (x) < 0$ .
What is a stationary point?
A stationary point is a point on a function where $f^{'} (x) = 0$ .
I.e., it is a point on a function where the tangent is horizontal.
What is a normal (or normal line) with regard to a function?
A normal (or normal line) is a straight line that passes through a point on a curve and is perpendicular to the tangent at that point.
What is the relationship between the gradients of a tangent and a normal at the same point on a curve?
At the same point on a curve, the product of the gradients of a tangent and its normal is equal to -1.
True or False?
Local minimum and maximum points are types of stationary points.
True.
Local minimum and maximum points are types of stationary points.
What is a local minimum point?
A local minimum point is a point where the function value is the lowest in the immediate vicinity.
True or False.
The equation of the tangent to $y = f (x)$ at the point $(x_{1}, y_{1})$ can be found by using $y - y_{1} = f (x_{1}) (x - x_{1})$ .
False.
To find the equation of the tangent you need to use the value of the derivative (gradient function) of the curve at the point $(x_{1}, y_{1})$ .
The equation of the tangent to $y = f (x)$ at the point $(x_{1}, y_{1})$ can be found by using $y - y_{1} = f^{'} (x_{1}) (x - x_{1})$ .
True or False.
A local maximum point is the point at which a function takes on its maximum value.
False.
A local maximum point is not necessarily the point at which a function takes on its maximum value.
A local maximum point is a point at which the function value is the highest in the immediate vicinity. There may however be other points (not in the immediate vicinity) at which the function takes on a higher value.
How can you use gradients to determine if a stationary point is a local minimum or local maximum?
To determine if a stationary point is a local minimum or local maximum, look at the gradient of the function on either side of the stationary point.
- If the gradient is positive to the left of the point and negative to the right, then the stationary point is a local maximum.
- If the gradient is negative to the left of the point and positive to the right, then the stationary point is a local minimum.
What is an equation for the normal to a curve $y = f (x)$ at the point $(x_{1}, y_{1})$ ?
An equation for the normal to a curve $y = f (x)$ at the point $(x_{1}, y_{1})$ is $y - y_{1} = - \frac{1}{f^{'} (x_{1})} (x - x_{1})$ .
What are optimisation problems?
Optimisation problems are problems that involve maximizing or minimising a quantity.
What is a constraint?
A constraint is a condition that limits the possible values of variables in a problem.
True or False?
A useful first step in solving an optimisation problem is rewriting the quantity to be optimised as a function of a single variable.
True.
A useful first step in solving an optimisation problem is rewriting the quantity to be optimised as a function of a single variable.
True or False?
A GDC can be used to find local maximum or minimum points in optimisation problems.
True.
A GDC can be used to find local maximum or minimum points in optimisation problems.
One way to do this is to use the GDC's graphing features to draw a graph of the function.
True or False?
The context of a question is not important when writing down the final solution of an optimisation problem.
False.
The final step in solving an optimisation problem is interpreting the answer in the context of the question.
True or False?
Optimisation problems always have a unique solution.
False.
Optimisation problems may have multiple solutions or no solution.
Why is it important to interpret the solution to an optimisation problem in the context of the problem?
Interpreting the solution to an optimisation problem in the context of the problem is important to ensure the answer makes sense in the real-world situation being modelled.

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Differentiation (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