11X1 T08 02 first principles
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Transcript of 11X1 T08 02 first principles
![Page 1: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/1.jpg)
The Slope of a Tangent to a Curve
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The Slope of a Tangent to a Curvey
x
y f x
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The Slope of a Tangent to a Curvey
x
P
y f x
k
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The Slope of a Tangent to a Curvey
x
P
y f x
k
Q
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The Slope of a Tangent to a CurveSlope PQ is an estimate for the slope of line k.
y
x
P
y f x
k
Q
![Page 6: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/6.jpg)
The Slope of a Tangent to a CurveSlope PQ is an estimate for the slope of line k.
y
x
P
y f x
k
Q
Q: Where do we position Q to get the best estimate?
![Page 7: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/7.jpg)
The Slope of a Tangent to a CurveSlope PQ is an estimate for the slope of line k.
y
x
P
y f x
k
Q
Q: Where do we position Q to get the best estimate?
A: As close to P as possible.
![Page 8: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/8.jpg)
The Slope of a Tangent to a CurveSlope PQ is an estimate for the slope of line k.
y
x
P
y f x
k
Q
Q: Where do we position Q to get the best estimate?
A: As close to P as possible.
P
Q
![Page 9: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/9.jpg)
The Slope of a Tangent to a CurveSlope PQ is an estimate for the slope of line k.
y
x
P
y f x
k
Q
Q: Where do we position Q to get the best estimate?
A: As close to P as possible.
P
Q
,x f x
![Page 10: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/10.jpg)
The Slope of a Tangent to a CurveSlope PQ is an estimate for the slope of line k.
y
x
P
y f x
k
Q
Q: Where do we position Q to get the best estimate?
A: As close to P as possible.
P
Q
,x f x
,x h f x h
![Page 11: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/11.jpg)
The Slope of a Tangent to a CurveSlope PQ is an estimate for the slope of line k.
y
x
P
y f x
k
Q
Q: Where do we position Q to get the best estimate?
A: As close to P as possible.
P
Q
,x f x
,x h f x h
PQf x h f x
mx h x
f x h f xh
![Page 12: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/12.jpg)
The Slope of a Tangent to a CurveSlope PQ is an estimate for the slope of line k.
y
x
P
y f x
k
Q
Q: Where do we position Q to get the best estimate?
A: As close to P as possible.
P
Q
,x f x
,x h f x h
PQf x h f x
mx h x
f x h f xh
To find the exact value of the slope of k, we calculate the limit of the slope PQ as h gets closer to 0.
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0
slope of tangent = limh
f x h f xh
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0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised;
![Page 15: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/15.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx
![Page 16: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/16.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y
![Page 17: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/17.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x
![Page 18: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/18.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
![Page 19: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/19.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx the derivative measures the rate
of something changing
![Page 20: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/20.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
the derivative measures the rate
of something changing
![Page 21: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/21.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
The process is called “differentiating from first principles”
the derivative measures the rate
of something changing
![Page 22: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/22.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
The process is called “differentiating from first principles” e.g. Differentiate 6 1 by using first principles.i y x
the derivative measures the rate
of something changing
![Page 23: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/23.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
The process is called “differentiating from first principles” e.g. Differentiate 6 1 by using first principles.i y x
6 1f x x
the derivative measures the rate
of something changing
![Page 24: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/24.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
The process is called “differentiating from first principles” e.g. Differentiate 6 1 by using first principles.i y x
6 1f x x
6 16 6 1
f x h x hx h
the derivative measures the rate
of something changing
![Page 25: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/25.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
The process is called “differentiating from first principles” e.g. Differentiate 6 1 by using first principles.i y x
6 1f x x
6 16 6 1
f x h x hx h
0
limh
f x h f xdydx h
the derivative measures the rate
of something changing
![Page 26: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/26.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
The process is called “differentiating from first principles” e.g. Differentiate 6 1 by using first principles.i y x
6 1f x x
6 16 6 1
f x h x hx h
0
limh
f x h f xdydx h
0
6 6 1 6 1limh
x h xh
the derivative measures the rate
of something changing
![Page 27: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/27.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
The process is called “differentiating from first principles” e.g. Differentiate 6 1 by using first principles.i y x
6 1f x x
6 16 6 1
f x h x hx h
0
limh
f x h f xdydx h
0
6 6 1 6 1limh
x h xh
0
6limh
hh
the derivative measures the rate
of something changing
![Page 28: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/28.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
The process is called “differentiating from first principles” e.g. Differentiate 6 1 by using first principles.i y x
6 1f x x
6 16 6 1
f x h x hx h
0
limh
f x h f xdydx h
0
6 6 1 6 1limh
x h xh
0
6limh
hh
0lim6h
the derivative measures the rate
of something changing
![Page 29: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/29.jpg)
0
slope of tangent = limh
f x h f xh
This is known as the “derivative of y with respect to x” and is symbolised; dy
dx, y , f x , d f x
dx
0
limh
f x h f xf x
h
The process is called “differentiating from first principles” e.g. Differentiate 6 1 by using first principles.i y x
6 1f x x
6 16 6 1
f x h x hx h
0
limh
f x h f xdydx h
0
6 6 1 6 1limh
x h xh
0
6limh
hh
0lim6h
6
the derivative measures the rate
of something changing
![Page 30: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/30.jpg)
2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
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2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x
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2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x 2 5 2f x h x h x h
![Page 33: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/33.jpg)
2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x 2 5 2f x h x h x h
2 22 5 5 2x xh h x h
![Page 34: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/34.jpg)
2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x 2 5 2f x h x h x h
2 22 5 5 2x xh h x h
0
limh
f x h f xdydx h
![Page 35: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/35.jpg)
2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x 2 5 2f x h x h x h
2 22 5 5 2x xh h x h
0
limh
f x h f xdydx h
2 2 2
0
2 5 5 2 5 2limh
x xh h x h x xh
![Page 36: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/36.jpg)
2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x 2 5 2f x h x h x h
2 22 5 5 2x xh h x h
0
limh
f x h f xdydx h
2 2 2
0
2 5 5 2 5 2limh
x xh h x h x xh
2
0
2 5limh
xh h hh
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2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x 2 5 2f x h x h x h
2 22 5 5 2x xh h x h
0
limh
f x h f xdydx h
2 2 2
0
2 5 5 2 5 2limh
x xh h x h x xh
2
0
2 5limh
xh h hh
0lim 2 5h
x h
![Page 38: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/38.jpg)
2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x 2 5 2f x h x h x h
2 22 5 5 2x xh h x h
0
limh
f x h f xdydx h
2 2 2
0
2 5 5 2 5 2limh
x xh h x h x xh
2
0
2 5limh
xh h hh
0lim 2 5h
x h
2 5x
![Page 39: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/39.jpg)
2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x 2 5 2f x h x h x h
2 22 5 5 2x xh h x h
0
limh
f x h f xdydx h
2 2 2
0
2 5 5 2 5 2limh
x xh h x h x xh
2
0
2 5limh
xh h hh
0lim 2 5h
x h
2 5x
when 1, 2 1 5
3
dyxdx
![Page 40: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/40.jpg)
2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
2 5 2f x x x 2 5 2f x h x h x h
2 22 5 5 2x xh h x h
0
limh
f x h f xdydx h
2 2 2
0
2 5 5 2 5 2limh
x xh h x h x xh
2
0
2 5limh
xh h hh
0lim 2 5h
x h
2 5x
when 1, 2 1 5
3
dyxdx
the slope of the tangent at 1, 2 is 3
![Page 41: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/41.jpg)
2 3 1y x
![Page 42: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/42.jpg)
2 3 1y x
2 3 33 1
y xy x
![Page 43: 11X1 T08 02 first principles](https://reader033.fdocuments.us/reader033/viewer/2022052906/55895b44d8b42a81178b469d/html5/thumbnails/43.jpg)
2 3 1y x
2 3 33 1
y xy x
Exercise 7B; 1, 2adgi, 3(not iv), 4, 7ab i,v, 12 (just h approaches 0)