Download - 11X1 T09 02 first principles (2011)
The Slope of a Tangent to a Curve
The Slope of a Tangent to a Curvey
x
y f x
The Slope of a Tangent to a Curvey
x
P
y f x
k
The Slope of a Tangent to a Curvey
x
P
y f x
k
Q
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
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?
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.
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
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
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
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
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.
0
slope of tangent = limh
f x h f xh
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;
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
2 Find the equation of the tangent to 5 2 at the point 1, 2 .ii y x x
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 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 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
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 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 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
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 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
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
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
2 3 1y x
2 3 1y x
2 3 33 1
y xy x
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)
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