Define the term gradient .

Gradient is a measure of steepness , referring to how steep a line or curve is on a graph. It describes the rate at which y changes with respect to x .

True or False? The gradient of a curve is always constant .

False. The gradient of a straight line is constant . The gradient of a curve changes as the value of x changes.

Define the term tangent in the context of a curve on a graph.

In the context of a curve on a graph, a tangent is a straight line that touches the curve at one point (without otherwise cutting across the curve near that point).

What is the gradient function ?

The gradient function , also known as the derivative or derived function , is an algebraic function that takes inputs of x -coordinates and gives outputs of gradients at the points with those x -coordinates.

Define differentiation .

Differentiation is the operation that turns curve equations into gradient functions .

What is the rule for differentiating a term like 10 x ?

When differentiating a term like 10 x (where there is no power written next to the x ), remember that this means 10x 1 , so will differentiate to 1×10x 0 , which is 10.

True or False? Any constant term (number on its own) differentiates to zero .

True. Any constant term (number on its own) differentiates to zero .

What are the steps to find the gradient of a curve at a particular point using the gradient function ?

To find the gradient of a curve at a particular point using the gradient function : Find the x -coordinate of the point on the curve Use differentiation to find the gradient function for the curve Substitute the x -coordinate into the gradient function

What is the definition of a stationary point ?

A stationary point is a point at which the gradient of a curve is zero .

What is the definition of a turning point ?

A turning point is a point at which a curve changes from moving upwards to moving downwards, or vice versa. I.e. it is a maximum (peak) or minimum (trough) on the curve.

What condition does the gradient of a curve satisfy at a turning point ?

At a turning point , the gradient of a curve is equal to zero .

True or false? The term 'stationary point' is also used for turning points.

True. Turning points are also referred to as stationary points . A stationary point is any point on a curve where the gradient is equal to zero.

What are the steps to find the coordinates of a turning point ?

To find the coordinates of a turning point : Set the derivative (gradient function) equal to zero and solve to find the x -coordinate Substitute the x -coordinate into the equation of the curve to find the y -coordinate (In step 2, be sure to use the original equation of the curve, not the gradient function!)

True or false? After using a gradient function to find the x -coordinate of a turning point , you should substitute that x -coordinate back into the gradient function to find the y -coordinate of the turning point .

False. After using a gradient function to find the x -coordinate of a turning point, you should substitute that x -coordinate back into the original equation of the curve to find the y -coordinate of the turning point.

Define a maximum point .

A maximum point is a type of stationary point (or turning point) where a graph reaches the top of a "peak" . (It is sometimes called a local maximum point, because there may be other parts of the graph that reach higher values.)

Define a minimum point .

A minimum point is a type of stationary point (or turning point) where the graph reaches the bottom of a "trough" . (It is sometimes called a local minimum point, because there may be other parts of the graph that reach lower values.)

How can you use the gradient function (derivative) to classify a turning point?

At a maximum point : the gradient just before the turning point is positive the gradient at the turning point is zero the gradient just after the turning point is negative At a minimum point : the gradient just before the turning point is negative the gradient at the turning point is zero the gradient just after the turning point is positive

True or False? You can use differentiation to find the maximum or minimum values of real-life problems, such as the area of a field and the volume of water in a lake.

True. You can use differentiation to find the maximum or minimum values of real-life problems, such as the area of a field and the volume of water in a lake.

What is a 'particle' in a kinematics question?

In kinematics questions, the object that is moving is usually referred to as a 'particle' . It is modelled as taking up a single point in space.

Define displacement .

Displacement is a vector quantity that measures how far a particle is from a fixed origin point. It can be positive or negative, and is usually measured in metres (m).

What is the difference between displacement and distance ?

Distance only measures 'how far' , and is always a positive number. Displacement measures 'how far and in what direction' , so in kinematics questions it can be positive or negative.

Velocity is a vector quantity that measures how fast a particle is traveling. It can be positive or negative, and is usually measured in metres per second (m/s).

What is the difference between velocity and speed ?

Speed only measures 'how fast' , and is always a positive number. Velocity measures 'how fast and in what direction' , so in kinematics questions it can be positive or negative.

What does a zero velocity value indicate?

If velocity is zero , then the particle is stationary (not moving).

Define acceleration .

Acceleration is a vector quantity that measures how fast the velocity is changing , It can be positive or negative, and is usually measured in metres per second squared (m/s 2 ).

What does a zero acceleration value indicate?

If acceleration is zero , then the particle is moving at a constant velocity (i.e., its velocity is not changing).

What does "initial" or "initially" mean in the context of a kinematics question?

"Initial" or "initially" refers to the conditions when t = 0 (at the start).

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Define the term gradient.
Gradient is a measure of steepness, referring to how steep a line or curve is on a graph. It describes the rate at which y changes with respect to x.
True or False?
The gradient of a curve is always constant.
False.
The gradient of a straight line is constant.
The gradient of a curve changes as the value of x changes.
Define the term tangent in the context of a curve on a graph.
In the context of a curve on a graph, a tangent is a straight line that touches the curve at one point (without otherwise cutting across the curve near that point).
What is the gradient function?
The gradient function, also known as the derivative or derived function, is an algebraic function that takes inputs of x-coordinates and gives outputs of gradients at the points with those x-coordinates.
What does $\frac{d y}{d x}$ represent?
$\frac{d y}{d x}$ represents the gradient function or derivative. It is pronounced "dy by dx" or "dy over dx".
Define differentiation.
Differentiation is the operation that turns curve equations into gradient functions.
What is the main rule for differentiating terms with powers of $x$ ?
To differentiate powers of $x$ :
- multiply the number in front by the power
- then subtract one from the power
So the derivative of $2 x^{3}$ is $(3 \times 2) x^{3 - 1} = 6 x^{2}$ .
What is the rule for differentiating a term like 10x?
When differentiating a term like 10x (where there is no power written next to the x), remember that this means 10x¹, so will differentiate to 1×10x⁰, which is 10.
True or False?
Any constant term (number on its own) differentiates to zero.
True.
Any constant term (number on its own) differentiates to zero.
What is the main rule for differentiating a number of terms added or subtracted together? (For example something like $2 x^{3} - 3 x + 4$ .)
The main rule for differentiating a number of terms added or subtracted together is simply to differentiate each term one at a time.
So the derivative of $2 x^{3} - 3 x + 4$ is $(3 \times 2) x^{3 - 1} - 3 + 0 = 6 x^{2} - 3$ .
What are the steps to find the gradient of a curve at a particular point using the gradient function?
To find the gradient of a curve at a particular point using the gradient function:
1. Find the x-coordinate of the point on the curve
2. Use differentiation to find the gradient function for the curve
3. Substitute the x-coordinate into the gradient function
What is the definition of a stationary point?
A stationary point is a point at which the gradient of a curve is zero.
What is the definition of a turning point?
A turning point is a point at which a curve changes from moving upwards to moving downwards, or vice versa.
I.e. it is a maximum (peak) or minimum (trough) on the curve.
What condition does the gradient of a curve satisfy at a turning point?
At a turning point, the gradient of a curve is equal to zero.
State an equation involving the derivative that can be used for finding the x-coordinate of a turning point.
At a turning point the derivative is equal to zero, so an equation that can be used to find the x-coordinate of a turning point is $\frac{d y}{d x} = 0$ .
(To find the x-coordinate(s) of the turning point(s), solve that equation for x.)
True or false?
The term 'stationary point' is also used for turning points.
True.
Turning points are also referred to as stationary points. A stationary point is any point on a curve where the gradient is equal to zero.
What are the steps to find the coordinates of a turning point?
To find the coordinates of a turning point:
1. Set the derivative (gradient function) equal to zero and solve to find the x-coordinate
2. Substitute the x-coordinate into the equation of the curve to find the y-coordinate
(In step 2, be sure to use the original equation of the curve, not the gradient function!)
True or false?
After using a gradient function to find the x-coordinate of a turning point, you should substitute that x-coordinate back into the gradient function to find the y-coordinate of the turning point.
False.
After using a gradient function to find the x-coordinate of a turning point, you should substitute that x-coordinate back into the original equation of the curve to find the y-coordinate of the turning point.
Define a maximum point.
A maximum point is a type of stationary point (or turning point) where a graph reaches the top of a "peak".
(It is sometimes called a local maximum point, because there may be other parts of the graph that reach higher values.)
Define a minimum point.
A minimum point is a type of stationary point (or turning point) where the graph reaches the bottom of a "trough".
(It is sometimes called a local minimum point, because there may be other parts of the graph that reach lower values.)
True or False?
A positive parabola (positive $x^{2}$ term) always has exactly one maximum point.
False.
A positive parabola always has exactly one minimum point (and no maximum points).
True or False?
A negative parabola (negative $x^{2}$ term) always has exactly one maximum point.
True.
A negative parabola always has exactly one maximum point (and no minimum points).
True or False?
A positive cubic curve (positive $x^{3}$ term) has a maximum point on the left and a minimum point on the right.
True.
A positive cubic curve (positive $x^{3}$ term) has a maximum point on the left and a minimum point on the right.
True or False?
A negative cubic curve (negative $x^{3}$ term) has a maximum point on the left and a minimum point on the right.
False.
A negative cubic curve (negative $x^{3}$ term) has a minimum point on the left and a maximum point on the right.
How can you use the gradient function (derivative) to classify a turning point?
At a maximum point:
- the gradient just before the turning point is positive
- the gradient at the turning point is zero
- the gradient just after the turning point is negative
At a minimum point:
- the gradient just before the turning point is negative
- the gradient at the turning point is zero
- the gradient just after the turning point is positive
True or False?
You can use differentiation to find the maximum or minimum values of real-life problems, such as the area of a field and the volume of water in a lake.
True.
You can use differentiation to find the maximum or minimum values of real-life problems, such as the area of a field and the volume of water in a lake.
Without doing any calculations, how do you know that the graph of $y = x^{2} - 10 x + 26$ has a minimum point and not a maximum point?
The graph of $y = x^{2} - 10 x + 26$ has a positive $x^{2}$ term so it has the shape of a positive parabola (quadratic curve).
This means it is a U shaped curve, so must have a minimum point (not a maximum point).
The area of a shape, $A$ m², is given by $A = x^{2} - 6 x + 14$ .
Explain how you would use differentiation to find the minimum area.
The area of a shape, $A$ m², is given by $A = x^{2} - 6 x + 14$ .
To find the minimum area
1. Differentiate $A$ to get $\frac{d A}{d x}$
2. Set $\frac{d A}{d x} = 0$ to form and solve an equation in $x$
3. Substitute this value of $x$ back into $A$
4. This must be the minimum value of $A$ as $A = x^{2} - 6 x + 14$ is a positive quadratic curve (U-shaped)
True or False?
You can find the maximum or minimum value of $A = x^{2} + 2 y$ by setting $\frac{d A}{d x} = 0$ .
False.
You cannot find the maximum or minimum value of $A = x^{2} + 2 y$ by setting $\frac{d A}{d x} = 0$ because $A$ needs to be in terms of one variable only.
You cannot have both $x$ and $y$ in the equation you want to maximise / minimise.
A rectangular field measures $x$ m by $y$ m with a perimeter of $100$ m.
Explain how you would find its area, $A$ m², in terms of $x$ only.
A rectangular field measures $x$ m by $y$ m with a perimeter of $100$ m.
To find its area, $A$ m², in terms of $x$ only
1. First find $A$ in terms of $x$ and $y$
  ( $A = x y$ )
2. Then use the perimeter information to find a relationship between $x$ and $y$
  ( $2 x + 2 y = 100$ )
3. Make $y$ the subject ( $y = 50 - x$ ) and substitute it into $A$
This gives $A = x (50 - x)$ which expands to give $A = 50 x - x^{2}$ . This is now in one variable only.
True or False?
The minimum volume of the shape given by $V = x^{3} - 12 x + 20$ is the value of $x$ that satisfies $\frac{d V}{d x} = 0$ .
False.
The minimum volume of the shape given by $V = x^{3} - 12 x + 20$ is not the value of $x$ that satisfies $\frac{d V}{d x} = 0$ as the minimum volume will be a value of $V$ , not $x$ .
You need to find the two values of $x$ that satisfies $\frac{d V}{d x} = 0$ then substitute them into the equation for $V$ to get a maximum and a minimum value. The smaller value is the minimum value of $V$ required.
What is a 'particle' in a kinematics question?
In kinematics questions, the object that is moving is usually referred to as a 'particle'. It is modelled as taking up a single point in space.
Define displacement.
Displacement is a vector quantity that measures how far a particle is from a fixed origin point.
It can be positive or negative, and is usually measured in metres (m).
What is the difference between displacement and distance?
Distance only measures 'how far', and is always a positive number.
Displacement measures 'how far and in what direction', so in kinematics questions it can be positive or negative.
What letter is usually used for displacement in kinematics questions?
Displacement is usually indicated by the letter $s$ in kinematics questions.
So the displacement of a particle at time $t$ , for example, might be given as $s = 3 t^{2} - 2 t + 1$ .
Define velocity.
Velocity is a vector quantity that measures how fast a particle is traveling.
It can be positive or negative, and is usually measured in metres per second (m/s).
What is the difference between velocity and speed?
Speed only measures 'how fast', and is always a positive number.
Velocity measures 'how fast and in what direction', so in kinematics questions it can be positive or negative.
If you know the equation for the displacement of a particle, how can you use calculus to find the velocity?
Velocity is the derivative of displacement, so if you know the displacement equation for a particle you can differentiate it to find the the velocity.
As an equation, that is $v = \frac{d s}{d t}$ .
What does a zero velocity value indicate?
If velocity is zero, then the particle is stationary (not moving).
Define acceleration.
Acceleration is a vector quantity that measures how fast the velocity is changing,
It can be positive or negative, and is usually measured in metres per second squared (m/s²).
If you know the equation for the velocity of a particle, how can you use calculus to find the acceleration?
Acceleration is the derivative of velocity, so if you know the velocity equation for a particle you can differentiate it to find the the acceleration.
As an equation, that is $a = \frac{d v}{d t}$ .
What does a zero acceleration value indicate?
If acceleration is zero, then the particle is moving at a constant velocity (i.e., its velocity is not changing).
What does "initial" or "initially" mean in the context of a kinematics question?
"Initial" or "initially" refers to the conditions when t = 0 (at the start).

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1. Topic 1
  Vectors
2. Topic 2
  Transformations

Differentiation (Edexcel IGCSE Maths A)

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