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Mathematics – Grade 12 Notes
Calculus – Gradients and Derivatives
Content:
Gradients and Derivatives:
1.1 Important Definitions and Notes for Gradients
1.2 Important Definitions and Notes for Average Gradients
1.3 Gradient from First Principles
1.4 Rules for Differentiation
1.1 Important Definitions and Notes for Gradients
➢ Gradient/Slope – The gradient or slope of a function indicates the rate at which 𝑦
changes as 𝑥 changes. Furthermore, it indicates the ‘steepness’ (inclination) of the
graph compared with the 𝑥-axis.
➢ Important Notations:
o The gradient of a graph is often symbolised by 𝑚 or 𝑎.
o The change in a variable is symbolised by ∆.
∆𝑦
▪ Thus, the gradient is equal to the change in 𝑦 over the change in 𝑥 (∆𝑥).
➢ The straight line (linear) graph is the only graph that has a constant gradient. In other
words, the gradient does not change. At every point along the graph, the gradient is
the same.
∆𝑦 𝑦2 − 𝑦1
𝑚= =
∆𝑥 𝑥2 − 𝑥1
The gradient is found by subtracting the
two 𝑦-values from each other and then
dividing that by the subtraction of the
two 𝑥-values but it very important to
keep the same order when subtracting.
As seen above it is point 1 (𝑥1 , 𝑦1 ) that is
subtracted from point 2 (𝑥2 , 𝑦2 ).
Pia Eklund 2023
, 2
➢ Unlike straight line graphs, curves have a constantly changing gradient. Let’s look at
the parabola as an example of this:
Let’s observe the gradient of the
parabola from left to right.
As we can see, the gradient starts of
very steep and has a negative
gradient (as 𝑥 increases in value, 𝑦
decreases). As it gets closer and
closer to the turning point (tip) the
gradient becomes less and less steep.
At the turning point the gradient is 0
(there is no angle between the graph
and the 𝑥-axis). After the turning point
it continues in the positive direction (as
𝑥 increases, 𝑦 also increases) and
becomes steeper and steeper.
1.2 Important Definitions and Notes for Average Gradients
➢ The average gradient between two points on a curve can be found by joining the two
points with a straight line and then finding the gradient of that line.
∆𝑦 𝑦𝑏 −𝑦𝑎
Average gradient = 𝑚 = =
∆𝑥 𝑥𝑏 −𝑥𝑎
➢ Secant – a line that passes through any two distinct point on a curve.
➢ Tangent – a line that touches the curve at one point.
https://www.desmos.com/calculator
Pia Eklund 2023
Mathematics – Grade 12 Notes
Calculus – Gradients and Derivatives
Content:
Gradients and Derivatives:
1.1 Important Definitions and Notes for Gradients
1.2 Important Definitions and Notes for Average Gradients
1.3 Gradient from First Principles
1.4 Rules for Differentiation
1.1 Important Definitions and Notes for Gradients
➢ Gradient/Slope – The gradient or slope of a function indicates the rate at which 𝑦
changes as 𝑥 changes. Furthermore, it indicates the ‘steepness’ (inclination) of the
graph compared with the 𝑥-axis.
➢ Important Notations:
o The gradient of a graph is often symbolised by 𝑚 or 𝑎.
o The change in a variable is symbolised by ∆.
∆𝑦
▪ Thus, the gradient is equal to the change in 𝑦 over the change in 𝑥 (∆𝑥).
➢ The straight line (linear) graph is the only graph that has a constant gradient. In other
words, the gradient does not change. At every point along the graph, the gradient is
the same.
∆𝑦 𝑦2 − 𝑦1
𝑚= =
∆𝑥 𝑥2 − 𝑥1
The gradient is found by subtracting the
two 𝑦-values from each other and then
dividing that by the subtraction of the
two 𝑥-values but it very important to
keep the same order when subtracting.
As seen above it is point 1 (𝑥1 , 𝑦1 ) that is
subtracted from point 2 (𝑥2 , 𝑦2 ).
Pia Eklund 2023
, 2
➢ Unlike straight line graphs, curves have a constantly changing gradient. Let’s look at
the parabola as an example of this:
Let’s observe the gradient of the
parabola from left to right.
As we can see, the gradient starts of
very steep and has a negative
gradient (as 𝑥 increases in value, 𝑦
decreases). As it gets closer and
closer to the turning point (tip) the
gradient becomes less and less steep.
At the turning point the gradient is 0
(there is no angle between the graph
and the 𝑥-axis). After the turning point
it continues in the positive direction (as
𝑥 increases, 𝑦 also increases) and
becomes steeper and steeper.
1.2 Important Definitions and Notes for Average Gradients
➢ The average gradient between two points on a curve can be found by joining the two
points with a straight line and then finding the gradient of that line.
∆𝑦 𝑦𝑏 −𝑦𝑎
Average gradient = 𝑚 = =
∆𝑥 𝑥𝑏 −𝑥𝑎
➢ Secant – a line that passes through any two distinct point on a curve.
➢ Tangent – a line that touches the curve at one point.
https://www.desmos.com/calculator
Pia Eklund 2023
