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

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

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.

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…

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

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

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

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)

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