FINDING THE SLOPE OF A CURVE AT A POINT:

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FINDING THE SLOPE OF A CURVE AT A POINT TAKING A
DERIVATIVE
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In order to find the slope of a curve at a point,
simply use the familiar formula for slope that
you learned in algebra or geometry and allow the
run to become smaller and smaller
?y
?x
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This technique works because any well-behaved
function, when viewed up close, approximates a
straight line, even though the function is not
really straight. For instance, even the surface
of the earth appears flat to someone standing on
its surface, even though it is really spherical.
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As ?x becomes smaller and smaller, this
approximation of the slope at the point will get
closer and closer to the true value. The value
that this approximation approaches (but never
actually reaches) is the slope of the curve at
that point. This technique may be used for any
value of x and f(x).
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Unfortunately, if we wanted to find the slope of
the curve at different points, we would have to
repeat this approximation technique at each
particular point where we wanted to find the
slope.
Is there a way that we could do this technique
just once and get a general result that will work
for any values of x and f(x)?
Yes, there is!
The slope of f(x) is another function, which we
will call m(x), or the slope function. If we
can find m(x), then we could simply plug in a
particular value of x to find the slope of the
curve at that particular value of x.
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Again, if we have a well-behaved function f(x),
then this functions slope is given by another
function, m(x). We need to find m(x).
m(x) ?
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We begin by using the formula for slope
We want to find m(x) for a given value of x. We
will call xi x and xf (x?x). Thus yi becomes
f(x) and yf becomes f(x?x).
We will also allow ?x to get closer and closer to
zero. In calculus terminology, we say that ?x
approaches zero, or ?x ? 0.
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f(x?x) f(x)
x x?x
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In calculus notation, we would write our
expression for m(x), the slope function, as
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In words we would say that m(x) is the limit of
our fraction as ?x approaches zero
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So how would we do this with a real function?
Let f(t) ½ t2. In this example, we are
allowing t, rather than x, to be our independent
variable, but the technique will still be the
same. If we replace x with t, we get
x becomes t x?x becomes t?t.
Also, f(x) becomes f(t) and f(x ?x)
becomes f(t?t).
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Since f(t) ½ t2, f(t?t) becomes ½ (t?t)2 ½
t2 2 t ?t (?t)2 Substituting all this into
our expression for m(t) gives
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We can now simplify our expression. Although ?t
approaches zero, it never actually equals zero.
Therefore, we can divide out one of the ?ts in
the third line
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But as ?t?0, ½ ?t also goes to zero. Thus our
final answer is
0
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In calculus notation, we say that our slope
function, m(t), is the derivative of f(t). In
this case, we say that the derivative of f(t)
with respect to t is t
Does our expression work? Lets see.
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Note that the dashed red line is the line tangent
to the curve y(t) ½ t2 when t 2.0 s.
The line tangent to a curve at a point is the
line that intersects the curve at that point and
is perpendicular to the curves radius of
curvature at that point.
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The equation of this tangent line is y 2t 2,
and it has a slope of 2 m/s.
y 2t 2 (slope is 2 m/s)
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This can be verified by calculating the slope
directly
y 2t 2 (slope is 2 m/s)
Slope Rise ?Run 6m ?3 s 2 m/s
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What about when t 4 s? According to our
derivative, the slope should be 4.0 m/s. Is this
correct?
Slope Rise ?Run 8 m ?2 s 4 m/s
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So we see that this technique of taking a
derivative does in fact give us a general
expression of the slope of the curve f(t) for any
value of t! In order to find the slope of a
curve at a particular value of t, simply insert
that value of t into your expression for the
curves derivative!