求解1. b
B
3
Investigation 1
When a curve is increasing, we say that it has a positive gradient, and
when it is decreasing we say that it has a negative gradient. Consider
the curve y=xº shown in the diagram.
1 a Give the range of x values for which the gradient of the curve is
i negative
I
ii positive
iii equal to zero.
b At which point is the gradient of the curve greater. A or B?
The tangent to a curve at a point is the straight line that just touches but does not
cross the curve at that point.
N
1
y=p
1
2.
3
2
2 a On your GDC or other software, plot the curve y=x' and draw the tangent
at the point (1, 1), as shown in the second diagram.
b Zoom in to the point (1, 1).
Factual What do you notice about the gradient of the tangent and the
gradient of the curve at the point of contact?
c Use your GDC or online software to draw tangents to y=x' at the points
listed below and in each case write down the gradient of the tangent.
0
-1
1
2
X
-1
0
2 2.
3
Gradient of tangent at x
2
d Conjecture a function for the gradient of y = x2 at the point on the curve with coordinates (x,x).
