fm calculus
TRANSCRIPT
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Starter
The line L passes through the points (0, 7) and (3, 19). Work out the equation of the line L.
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Starter
The line L passes through the points (0, 7) and (3, 19). Work out the equation of the line L.Gradient =
Equation: y=4x+cPasses through (0,7) so y-intercept is 7Therefore the equation is y=4x+7
How would we have worked out c if we were not given the y-intercept?
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Calculus - Differentiation
• Differentiation is a way of finding a gradient at a point on a curve.
• It is needed as curves have (by definition) a constantly changing gradient.
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Differentiation
• Why and how differentiation works is not required knowledge for the Further Maths exam
• It will not be covered in this session (look up Differentiation from First Principles if you want some light summer reading)
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Differentiation – How To…
OR:Multiply the whole thing by the power and reduce it by one
𝑦=𝑎𝑥𝑛❑⇒
𝑑𝑦𝑑𝑥=a 𝑥𝑛−1
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Differentiation - Example
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Differentiation – Try these
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Differentiation – Try these
𝟓 𝒙𝟒
𝟏𝟐𝒙𝟑
+6x
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Differentiation - Tangents
A tangent is a line that touches a curve at a single point.
The gradient of the tangent is equal to the gradient of the curve at that point.
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Differentiation - Tangents
As a straight line the equation of the tangent is:
y=mx + c
This is equal to evaluated at the point P.𝒅𝒚𝒅𝒙
You will need to know a specific point on the line to find c.
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Equations of Tangents
Find the equation of the tangent to , when x=2
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Equations of Tangents
Find the equation of the tangent to , when x=2 when x=2 => y=7x+c
Then when x=2, So the line passes through (2 , 3)
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Equations of Tangents
y=7x+c passing through (2,3)
So 3=7(2)+cc=-11
Equation is y=7x-11
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Equations of Tangents
Complete the table (only the tangents side)
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Tangents and Normals
• A normal is a line that is perpendicular to the tangent at a specific point.
• The gradient of a normal is the negative reciprocal of the tangent (-1/tangent gradient)
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Tangents and NormalsTangent Gradient Normal Gradient
4 -1/4
-3 1/3
1/2 -2
-1/3 3
3/4 -4/3
-7/2 2/7
Examples of negative reciprocals
Once you have worked out the gradient, finding the equation is exactly the same as for the tangent
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Tangents and Normals
Now complete the Normals side of the sheet