Developing the Formula for
Instantaneous Rates of Change
Using Limits
Learning Goal: To understand and develop the formula for
Instantaneous Rate of Change using limits.
Success Criteria: Students can…
• visually understand the concepts of secants, tangents, ARC and IRC
from a graph.
• develop and describe the notation and formulas needed to create the
IRC formula from a graph.
• apply the limit concept to develop, visualize and describe the IRC
formula.
• use technology to dynamically show the limit concept as applies to
IRC.
• Consider a function 𝑓 and a point 𝑃 𝑎, 𝑓(𝑎) . Choose a second, distinct
point on the graph called 𝑄 𝑎 + ℎ, 𝑓(𝑎 + ℎ) .
• For 𝑄 to be distinct from 𝑃, ℎ ≠ 0.
• We could have instead defined the second point as 𝑄 𝑏, 𝑓(𝑏) but this is the
version we want for defining derivatives in the next unit.
• We should always use the same notation for 𝑄, regardless whether 𝑄 follows or
precedes 𝑃. In MHF4U, you might have used 𝑄 𝑎 − ℎ, 𝑓(𝑎 − ℎ) when doing a
preceding interval. Do NOT do that any more. We want the SAME formula for
preceding and following intervals.
• Instead, we will let 𝒉 > 𝟎 when we want 𝑄 to follow 𝑃 and we will let 𝒉 < 𝟎
when we want 𝑄 to precede 𝑃.
• By this convention, 𝒉 represents the distance between the 𝑥 values of 𝑃 and 𝑄.
• The slope of secant 𝑃𝑄 (ARC) becomes 𝑚𝑃𝑄 =
𝑓 𝑎+ℎ −𝑓(𝑎)
𝑎+ℎ −𝑎
=
𝑓 𝑎+ℎ −𝑓(𝑎)
ℎ
• Let us now think conceptually about how a limit works.
lim
The thing we control:
We decide what this
thing approaches
The thing we are
interested in: What is
this thing approaching?
=
The answer to the
question: What is this
thing approaching?
• Let’s now look at how to apply these ideas to the diagram to get IRC.
lim
We want to
make point 𝑄
approach point 𝑃
We want to
make ℎ
approach 0.
We are interested in
the secant line: What is
it approaching?
We are actually
interested in the slope
of the secant line
(ARC): What is it
approaching?
=
The answer: The
tangent line
The answer: The slope
of the tangent line (IRC)
• We define the slope of the tangent (IRC) of a function 𝑓 at 𝑥 = 𝑎 to be:
𝑓 𝑎 + ℎ − 𝑓(𝑎)
𝑚 𝑇 = lim
ℎ→0
ℎ
• Recall: ℎ ≠ 0 is part of this limit by definition.
• By definition, this formula is an indeterminant form.
• Consider how this limit allows for 𝑄 to approach 𝑃 from both sides.