© annie patton implicit functions next slide. © annie patton aim of lesson next slide to...
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© Annie Patton
Implicit Functions
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© Annie Patton
Aim of Lesson
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To establish, what is an Implicit Function and how to differentiate one.
© Annie Patton
Implicit Functions
• In functions like y=x2 +2x +3, y is expressed in terms of x. This is called an Explicit Function.
• In functions like x2 +4xy –y2 =2x, y is not expressed in terms of x. This is called an Implicit Function.
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© Annie Patton
Differentiate x2 +4xy –y2 =2x with respect to x.
2 2d(x ) d(4xy) dy d(2x)+ - =
dx dx dx dx
2dy dx dy dy2x+4(x +y )- =2
dx dx dy dx
dy dy2x+4x +4y-2y =2
dx dx
dy(4x-2y)=2-2x-4y
dxdy 2-2x-4y
=dx 4x-2y
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© Annie Patton
Method
• Differentiate each term with respect to x.• Bring all the terms with to one side.• Find , in terms of x and y.
dy
dx
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dy
dx
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Find the equation of the tangent to the curve 3x2+y2=28 at the point (2,-4)
dyFirst weneedfind the slope or
dx2 2dx dy dy
3 + =0dx dy dx
6 2 0dy
x ydx
dy 6x 3x=- =-
dx 2y y
dy 6 3At the point (2,-4) =- =
dx -4 2
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Leaving Certificate Higher No 7(b) (1) Paper 1 2007
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Find the equation of the tangent to the curve 3x2+y2=28 at the point (2,-4) continued
3 3, .
2 2
dyAs the slope is equal todx
3( 4) ( 2)
2So the equation is y x
2( 4) 3( 2)y x
3 2 14 0x y
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–6 –4 –2 2 4 6
–10
–5
5
10
x
y
Start clicking when you want to see the answer.
© Annie Patton
2Find the slope of the tangent to the curve xy +y=6
at the point (1,2) .
2 6dxy dy d
dx dx dx
22 0
dy dx dyx ydx dx dx
22 0
dy dy dyx ydy dx dx
22dy dy
x y ydx dx
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5 10 15
–4
–3
–2
–1
1
2
3
4
x
yStart clicking when you want to see the answer.
2 4 4
2 1 4 1 5
the slope of the tangent
dy y
dx xy
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Leaving Certificate Higher No 7(b) (ii) Paper 1 2003
1 1 1 dyGiven + = , find the value of at the point (2,-3).
x y 6 dx
11 16
dd dyx
dx dx dx
1 1
0dx dy dy
dx dy dx
2 21 ( 1) 0dy
x ydx
2 2
1 1 dy
x y dx 2
2
9at (2,-3)=
4
dy y
dx x
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–10 10
–4
–3
–2
–1
1
2
3
4
x
y
Start clicking when you want to see the answer.
© Annie Patton
2 2x 2
Find the slope of the curve
y xy
2 2
0dx y dxy
dx dx
22 22 1 0dy dy
x y x x ydx dx
22 22 0dy dy dy
x xy y xdx dy dx
2 22 2 0dy dy
x xy y yxdx dx
2 2( 2 ) 2dy
x yx y xydx
2
2
2slope
2
dy y xythe
dx x yx
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Start clicking when you want to see the answer.
© Annie Patton
sin xDifferentiate y=
1-cos x
2dy dy dy=2y
dy dx dx
2
dy (1-cos x)(cos x)-sin x(0-(-sin x))2y =
dx (1-cos x)
2 2
2 2
dy cos x-cos x-sin x cos x-1 12y = = =-
dx (1-cos x) (1-cos x) 1-cos x
dy 1 1 1 1= = -
dx 2y 1-cosx 1-cosxsinx2
1-cosx
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2 sin
1 cos
xy
x
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© Annie Patton
A 20m ladder leans against a wall. The top slides down at the rate of 4 m/sec. How fast is the bottom of the ladder moving, when it is 16m from the wall?
4dy
dt
2 2 220x y 2 2
0
2 2 0
0
dx dy
dt dtdx dyx ydt dtdx dyx ydt dt
2 2
x=16
y= (20) (16) 12
When
16 12( 4) 0
3 / sec
dx
dtdx
mdt
20y
x
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Start clicking when you want to see the answer.
© Annie Patton
Implicit Functions
• In functions like y=x2 +2x +3, y is expressed in terms of x. This is called an Explicit Function.
• In functions like x2 +4xy –y2 =2x, y is not expressed in terms of x. This is called an Implicit Function.