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Tangents and Gradients
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Points with a Given Gradient
e.g. Find the coordinates of the points on the curve where the gradient equals 4 Gradient of curve = gradient of tangent = 4 We need to be able to find these points using algebra
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Points with a Given Gradient
e.g. Find the coordinates of the points on the curve where the gradient is 4 The gradient of the curve is given by Solution: Quadratic equation with no linear x -term Gradient is 4
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Points with a Given Gradient
The points on with gradient 4
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SUMMARY To find the point(s) on a curve with a given gradient: find the gradient function let equal the given gradient solve the resulting equation
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Exercises Find the coordinates of the points on the curves with the gradients given 1. where the gradient is -2 Ans: (-3, -6) 2. where the gradient is 3 ( Watch out for the common factor in the quadratic equation ) Ans: (-2, 2) and (4, -88)
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Increasing and Decreasing Functions
An increasing function is one whose gradient is always greater than or equal to zero. for all values of x A decreasing function has a gradient that is always negative or zero. for all values of x
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e.g.1 Show that is an increasing function
Solution: is the sum of a positive number ( 3 ) a perfect square ( which is positive or zero for all values of x, and a positive number ( 4 ) for all values of x so, is an increasing function
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e.g.2 Show that is an increasing function.
Solution: To show that is never negative ( in spite of the negative term ), we need to complete the square. for all values of x Since a square is always greater than or equal to zero, is an increasing function.
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The graphs of the increasing functions and are
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Exercises 1. Show that is a decreasing function and sketch its graph. 2. Show that is an increasing function and sketch its graph. Solutions are on the next 2 slides.
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Solutions 1. Show that is a decreasing function and sketch its graph. Solution: This is the product of a square which is always and a negative number, so for all x. Hence is a decreasing function.
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Solutions 2. Show that is an increasing function and sketch its graph. Solution: Completing the square: which is the sum of a square which is and a positive number. Hence y is an increasing function.
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(-1, 3) on line: The equation of a tangent
e.g. 1 Find the equation of the tangent at the point (-1, 3) on the curve with equation Solution: The gradient of a curve at a point and the gradient of the tangent at that point are equal (-1, 3) x Gradient = -5 At x = -1 (-1, 3) on line: So, the equation of the tangent is
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( We read this as “ f dashed x ” )
An Alternative Notation The notation for a function of x can be used instead of y. When is used, instead of using for the gradient function, we write ( We read this as “ f dashed x ” ) This notation is helpful if we need to substitute for x. e.g.
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e.g. 2 Find the equation of the tangent where x = 2 on the curve with equation where
Solution: To use we need to know y at the point as well as x and m From (1), (2, 2) on the line So, the equation of the tangent is
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SUMMARY To find the equation of the tangent at a point on the curve :
if the y-value at the point is not given, substitute the x -value into the equation of the curve to find y find the gradient function substitute the x-value into to find the gradient of the tangent, m substitute for y, m and x into to find c
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Exercises Find the equation of the tangent to the curve 1. at the point (2, -1) Ans: 2. Find the equation of the tangent to the curve at the point x = -1, where Ans:
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(-1, 3) on line: The equation of a tangent
x Solution: At x = -1 So, the equation of the tangent is Gradient = -5 (-1, 3) on line: e.g. 1 Find the equation of the tangent at the point (-1, 3) on the curve with equation The equation of a tangent
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Solution: To use we need to know y at the point as well as x and m
So, the equation of the tangent is From (1), (2, 2) on the line e.g. 2 Find the equation of the tangent where x = 2 on the curve with equation where
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