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this r ule. We are also familiar with the sign of the trigonometric functions in the
4 quadrants with the angle θ measured in the counter-clockwise (positive) direction. sin all tan cos Let y(x) be a function and let z1(x) be the tangent to the cur ve of y(x) at the
point x1. Let θ1 be the angle that the tangent z1(x) makes with the x-axis. From
the sign of tan θ1 we can gain more information about y(x) at x1. We can tell if y(x)
is increasing, decreasing, at an extremum (maximum or minimum), or changing
direction (flexing) at the point x1.
From the Analysis point of view, since the tangent z1(x) is a straight line, its
equation must be of the for m z1(x) = m1x + c1 , where m1 = tan θ1 is the
SLOPE of the tangent .
So, if we know θ1, then we may say that: z1(x) = tan θ1 . x + c1.
With a little Analysis we can show that FIRST DERIVATIVE y’(x1) = tan θ1.
So instead of geometrically :
1. drawing the graph of y(x) .
2. dr awing the tangent z1(x) to y(x) at some par ticular point y(x1) .
3. measuring the angle θ1 of the tangent z1(x) at y(x1) .
4. computing tan θ1
to tell the behaviour of y(x) at x1 , we may straight away use the FIRST DERIVATIVE
y’(x1) and analyse the behaviour of y(x) at x1.
96 We may apply this method of Analysis to two inter secting cur ves and analytically
compute the angle between two inter secting cur ves.
The most useful result of this par t is to be able to use the FIRST DERIVATIVE to find
the points at which y(x) is a maximum or minimum.
Another impor tant insight is:
if we know the tangents z1(x), z2(x), z3(x), . . . , zn (x), (fir st derivatives of y(x))
at sufficiently many points x1, x2, x3, . . . , xn , we may reconstr uct the cur ve y(x).
From the Analysis point of view, this insight tells us that we should be able to
calculate y(x) from y’(x). This inverse operation to find y(x) from its derivative
y’(x) is known as integration.
The reader may skip this par t on fir st reading a...
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- Fall '09
- Limit, Δx