Instantaneous Velocity & Limits | Calculus I
1. Average Velocity vs. Instantaneous Velocity
Average velocity
Average velocity over an interval :
Geometrically, this is the slope of the secant line through two points on .
Example: Usain Bolt
To estimate Bolt's velocity around seconds, use progressively smaller intervals around .
The lecture shows estimates such as:
On : approximately 12.28 m/s
On : approximately 12.27 m/s
On an even smaller interval: approximately 12.30 m/s
As the interval gets smaller, the average velocity gets closer to the velocity at exactly .
2. Instantaneous Velocity
The problem:
How do we determine velocity at one exact instant, rather than over an interval?
Average velocity only tells us what happens between two times.
To find velocity at , we let the second point get closer and closer to .
Secant → Tangent
A secant line passes through two points on a curve.
As the second point approaches :
The slope of the tangent line gives the instantaneous velocity.
Key idea
3. From Average Velocity to Instantaneous Velocity
Average velocity:
To get instantaneous velocity at , we make approach :
This is the mathematical formalization of the idea that the interval "gets close to" or "approaches" .
4. Limits
Definition — informal
means:
We can make as close as we want to by choosing sufficiently close to .
Important: approaches ; it doesn't necessarily have to equal .
5. Evaluating a Limit by Substitution
Example from the lecture:
Since this function behaves normally at , substitute:
Therefore:
The lecture notes that as gets sufficiently close to , approaches:
6. Why Can't We Always Just Plug In ?
Sometimes direct substitution works.
But sometimes something "wonky" happens at the limit point.
The most common example discussed is a rational function whose denominator approaches 0.
For example, directly plugging in might give:
which is undefined.
BUT:
A limit is concerned with what happens as approaches , not necessarily what happens at .
7. Holes in Graphs
A function can have a hole at .
That means:
There is a missing point in the graph
The function may be undefined at that exact point
But the function can still approach a definite value as
So you can have:
while simultaneously:
The lecture emphasizes that a missing value at does not automatically mean the limit doesn't exist.
HIGH-YIELD SUMMARY
Average velocity
→ slope of a secant line
Instantaneous velocity
→ slope of a tangent line
Limit
→ approaches as approaches
The conceptual progression
Average velocity
↓
Secant line
↓ smaller interval
Secant approaches tangent
↓
Instantaneous velocity
↓ mathematical formalization
Limit
Most important distinction
Function value:
= what the function actually equals at
Limit:
= what the function approaches as gets close to
They can be different, and the limit can exist even when is undefined.