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STROUD Worked examples and exercises are in the text PROGRAMME 8 DIFFERENTIATION APPLICATIONS
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PROGRAMME 8
DIFFERENTIATION APPLICATIONS
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Equation of a straight lineTangents and normals to a curve at a given pointMaximum and minimum valuesPoints of inflexion
Programme 8: Differentiation applications
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Equation of a straight lineTangents and normals to a curve at a given pointMaximum and minimum valuesPoints of inflexion
Programme 8: Differentiation applications
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Equation of a straight line (1)
Programme 8: Differentiation applications
The basic equation of a straight line is:
where: y mx c
gradient
intercept on the -axis
dymdx
c y
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Equation of a straight line
Programme 8: Differentiation applications
How about the equation of the line? Found it.
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Equation of a straight line (2)
Programme 8: Differentiation applications
Given the gradient m of a straight line and one point (x1, y1) through which it passes, the equation can be used in the form:
1 1( )y y m x x
Example:
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Equation of a straight line
Programme 8: Differentiation applications
Exercise 1
Found the equation of the straight line of:1. Line passing through (2, -3), gradient -2.2. Line passing through (5, 3), gradient 2.
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Programme 8: Differentiation applications Answers:1
2.
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Equation of a straight line (3)
Programme 8: Differentiation applications
If the gradient of a straight line is m and the gradient of a second straight line is m1 where the two lines are mutually perpendicular then:
1 111 that is mm mm
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Exercise A point P has coordinates (4,3) and the gradient m of straight line through P is 2. Then there is a line perpendicularly through P. Found the equation of the line.
Programme 8: Differentiation applications
Answer
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Programme 8: Differentiation applications
Are these two straight line perpendicular each other?
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Programme 8: Differentiation applications
Exercise 2
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Programme 8: Differentiation applications
Answers
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Programme 8: Differentiation applications
Further Example
1.
2.
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Programme 8: Differentiation applications
Answer (1)
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Programme 8: Differentiation applications
Answer (2)
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Equation of a straight lineTangents and normals to a curve at a given pointMaximum and minimum valuesPoints of inflexion
Programme 8: Differentiation applications
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Tangents and normals to a curve at a given pointTangent
Programme 8: Differentiation applications
The gradient of a curve, y = f (x), at a point P with coordinates (x1, y1) is given by the derivative of y (the gradient of the tangent) at the point:
The equation of the tangent can then be found from the equation:
1 1 at ( , )dy x ydx
1 1( ) where dyy y m x x mdx
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Tangents and normals to a curve at a given pointExample
Programme 8: Differentiation applications
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Tangents and normals to a curve at a given pointNormal
Programme 8: Differentiation applications
The gradient of a curve, y = f (x), at a point P with coordinates (x1, y1) is given by the derivative of y (the gradient of the tangent) at the point:
The equation of the normal (perpendicular to the tangent) can then be found from the equation:
1 1 at ( , )dy x ydx
1 11( ) where /
y y m x x mdy dx
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Tangents and normals to a curve at a given pointExample
Found the normal of the last exercise!
Programme 8: Differentiation applications
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Tangents and normals to a curve at a given pointExercise
Programme 8: Differentiation applications 1
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Equation of a straight lineTangents and normals to a curve at a given pointMaximum and minimum valuesPoints of inflexion
Programme 8: Differentiation applications 1
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Maximum and minimum values
Programme 9: Differentiation applications
A stationary point is a point on the graph of a function y = f (x) where the rate of change is zero. That is where:
This can occur at a local maximum, a local minimum or a point of inflexion. Solving this equation will locate the stationary points.
0dydx
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Maximum and minimum values
Programme 9: Differentiation applications
Having located a stationary point it is necessary to identify it. If, at the stationary point
2
2
2
2
0
the stationary point is a minimum
0
the stationary point is a maximum
d ydx
d ydx
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Maximum and minimum values
Programme 9: Differentiation applications
If, at the stationary point
The stationary point may be:
a local maximum, a local minimum or a point of inflexion
The test is to look at the values of y a little to the left and a little to the right of the stationary point
2
2 0d ydx
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ExampleProgramme 9: Differentiation applications
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Equation of a straight lineTangents and normals to a curve at a given pointMaximum and minimum valuesPoints of inflexion
Programme 8: Differentiation applications
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Points of inflexion
Programme 9: Differentiation applications
A point of inflexion can also occur at points other than stationary points. A point of inflexion is a point where the direction of bending changes – from a right-hand bend to a left-hand bend or vice versa.
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Points of inflexion
Programme 9: Differentiation applications
At a point of inflexion the second derivative is zero. However, the converse is not necessarily true because the second derivative can be zero at points other than points of inflexion.
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Points of inflexion
Programme 9: Differentiation applications
The test is the behaviour of the second derivative as we move through the point. If, at a point P on a curve:
and the sign of the second derivative changes as x increases from values to the left of P to values to the right of P, the point is a point of inflexion.
2
2 0 d ydx
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