Laboratory I.4
Explorations with the Slopes of Tangents
Goals
• Students will reinforce lecture concepts about the relationship between slopes of secant lines
and slopes of tangent lines.
• Students will understand why some functions do not have tangent lines at certain points.
Before the Lab
As you've seen in lecture, the slope of the tangent line depends on the limit of slopes of secant
lines. In this lab, we're going to ask you to draw secant lines. Here's some background algebra to
help you do it more efficiently in DERIVE.
The secant line goes through two points of the graph of a function
f
(
x
). The first point is the point
at which you're trying to find the derivative; let's call that point (
c
,
f
(
c
)). The second point is a
small distance
h
away: (
c
+
h
,
f
(
c
+
h
)). So the slope of the line between these two points is
h
c
f
h
c
f
c
h
c
c
f
h
c
f
x
y
)
(
)
(
)
(
)
(
)
(
−
+
=
−
+
−
+
=
Δ
Δ
and the pointslope formula for the secant line, y, with that slope through the point (
c
,
f
(
c
)) is
)
(
)
(
)
(
)
(
)
(
)
(
c
x
h
c
f
h
c
f
c
f
c
x
x
y
x
f
y
−
⋅
−
+
+
=
−
⋅
Δ
Δ
+
=
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Lab 4
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In the Lab (80 pts)
0. In this first section, we will set up some general formulas that will work with any function you
might care to use in DERIVE.
You will apply them in the later sections.
Author
f(x) :=
(yes, this is blank after the = sign)
This lets DERIVE know that the letter f will stand for a function, not an ordinary
variable.
Author
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 Spring '10
 Moody
 Derivative

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