1. 5.3 Definite Integrals,
Antiderivatives, and
the Average Value of
a Function

2. V = V(t) = velocity in feet/second
You’ve already approximated (using rectangles) the distance traveled by the object whose velocity is modeled in the graph to the left
t = time in seconds
Now find the actual distance traveled by the object over 4 seconds using the Definite Integral

3. Remember this from the fall? If you drive 100 miles north
…in 2 hours…
What was your average velocity for the trip?
100 miles
50 miles/hour
Does this mean that you were going 50 miles/hour the whole time?
No.
Were you at any time during the trip going 50 mi/hr?
Absolutely. There is no way that you couldn’t have been.

4. Now let’s look at average velocity from another perspective...
Suppose that we know for a fact that you were in fact going 50 mph the whole time.
t = time in hours
V = V(t) = velocity in miles/hour
Use your newfound skills to find the distance travelled over the 2 hour period using this graph.
To find the distance travelled…

5. Now let’s look at average velocity from another perspective...
Suppose that we know for a fact that you were in fact going 50 mph the whole time.
V = V(t) = velocity in miles/hour
Now use this to find the average velocity over those 2 hours.
t = time in hours

6. Now let’s look at average velocity from another perspective...
Suppose that we know for a fact that you were in fact going 50 mph the whole time.
mph
V = V(t) = velocity in miles/hour or
So if you were asked to find the average value of any function f(x) (that was continuous) over an interval [a,b], how would you do it?
t = time in hours

7. So if you were asked to find the average value of any function f(x) (that was continuous) over an interval [a,b], how would you do it? a b

8. Remember the original MVT?
If f is continuous on then at some point c in (a, b),
When looking at anti-derivatives and definite integrals, we write it another way:
Average Value Theorem (for definite integrals)
So we just say that:
Average Value of f (x)

9. V = V(t) = velocity in feet/second
Now find the average velocity of the object over 4 seconds using the Definite Integral
t = time in seconds