5.3 definite integrals, antiderivatives, and the average value of a function
TRANSCRIPT
5.3 Definite Integrals,Antiderivatives, and
the Average Value of a Function
211
8V t
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
V =
V(t
) =
vel
ocit
y in
fee
t/se
cond
Now find the actual distance traveled by the object over 4 seconds using the Definite Integral
4
0
2 18
1dtt
4
0
34
0
3
2438
1t
tt
t
0
24
04
24
4 33
feet3
26
If you drive 100 miles north
…in 2 hours…
What was your average velocity for the trip?
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.
100 miles
Remember this from the fall?
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
) =
vel
ocit
y in
mil
es/h
ourUse your newfound skills to
find the distance travelled over the 2 hour period using this graph.
50)( tv
To find the distance travelled…
2
0
2
050)( dtdttv
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
) =
vel
ocit
y in
mil
es/h
our
)0(50)2(50
tdt 50502
0
miles100
Now use this to find the average velocity over those 2 hours.
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
) =
vel
ocit
y in
mil
es/h
our
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?
502
100
hours
milesmph
hours
dx
2
502
0or
ab
dxxfb
a
)(
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?
)(xf
a b
Remember the original MVT?
Average Value Theorem (for definite integrals)
If f is continuous on then at some point c in (a, b), ,a b
1
b
af c f x dx
b a
ab
afbfcf
)()()(
When looking at anti-derivatives and definite integrals, we write it another way:
ab
aFbFcf
)()(
)(
ab
dxxfb
a
)(
So we just say that: Average Value of f (x)
211
8V t
t = time in seconds
V =
V(t
) =
vel
ocit
y in
fee
t/se
cond
Now find the average velocity of the object over 4 seconds using the Definite Integral
4
0
2 18
1
04
1dtt
4
0
3
244
1t
t
feet
3
26
sec4
1
sec/3
21 feet