The Fundamental Theorems of Calculus

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• The Fundamental Theorems of Calculus [5.5,FT] (.ppt)
• Handout (.pdf)

Thinking about distance and speed

Work through these questions from the handout together in your discussion groups.

Your odometer registers the number of miles your car has been driven since it was built. Your speedometer registers your speed at an instant.
WyldKyss

Suppose today at 7:00AM was 43797.0 hours after David's car was built. Suppose $f(t)$ is the miles on David's odometer $t$ hours after it was built. This statement says something about a trip that David took this morning: $$\frac{f(43797.5)-f(43797.0)}{43797.5-43797.0}=30.$$

1. What are the units of the numerator? miles
2. Make a reasonable guess about the units of the denominator. Looks like the same numbers as the "hours since the car was built", so hours
3. Calculate the denominator (carry out the subtraction) and interpret the denominator. 0.5, which means half an hour.
4. What are the units of 30? miles per hour or mph
5. What is your interpretation of this expression? This is the average speed of the car between 7:00 AM and 7:30 AM

This expression also says something about the trip: $$f'(43797.25)=50.$$

Give the units of the number 50 in this expression, and give an interpretation of what the expression means.

$f'=\frac{df}{dt}$, so the units for $f'$ is the same as the units of $f$ (miles) divided by the units of $t$ (hours), or $\frac{\text{miles}}{\text{hour}}$. $f'$ is the instantaneous rate of change. Also, 43797.25 hours is 0.25 hours after 43797.0, which was 7:00 AM. So, at 7:15 AM, the car was moving at a speed of 50 miles/hour. -OR- At 7:15 AM, David's speedometer read 50 mph.

Suppose $g(t)$ is the miles per hour on David's speedometer $t$ hours after it was built. This expression says something about the car trip this morning: $$g(43797.0)(0.2)+g(43797.2)(0.1)+g(43797.3)(0.2)=14.$$ [Hmmm, if this were a "Left sum" of rectangle areas, this might be approximating the area under the graph of what function? Between what starting value of the function's argument and what ending value?]

1. What are the units of $g$, the speedometer reading? miles per hour = miles / hours
2. Make a reasonable guess about the units of the numbers in parentheses: 0.2, 0.1, and 0.2.

   Hint: Oh, look:

   • 0.2 = $43797\color{red}.2\color{black} -43797\color{red}.0$ and
   • 0.1 = $43797\color{red}.3\color{black}-43797\color{red}.2$.

   Maybe hours? It looks like 0.2 and 0.1 are the differences between the number of hours in the arguments of $g$. And 0.2+0.1+0.2=0.5 which was the time for the trip given above.

3. In light of your answers above, what are the units of 14? Each product is (miles/hour)*(hours)=miles. So 14 miles.
4. What is your interpretation of this expression?

   I could invent a story:
   
   David glanced down at his speedometer at 7:00 am (=43797.0 hours), at 7:12, and at 7:18 during a half hour journey that lasted until 7:30. He estimated the distance that he drove during the first 12 minutes (12/60=0.2 hours) by multiplying his speed at 7 AM times 0.2 hours. He estimated a distance for each of the time intervals in the same way, and when he added those 3 estimates together, he came up with an estimate of his total distance travelled: 14 miles for his trip that lasted from 7:00 to 7:30.

Now consider this expression about the trip: $$\int_{43797.0}^{43797.5} g(t)\,dt=15.$$

1. Knowing the units of $g$ miles/hour and $dt$ hours, what are the units of the number 15? miles/hour * hours = miles,
2. Give an interpretation of the meaning of this expression.

   Imagine that David had not looked at his odometer when he started and stopped his trip, But did have his phone taking a video of his speedometer during his whole drive from 7:00 am to 7:30 am. He could have looked through the video frame-by-frame to have a function called $g(t)$ which was his speedometer reading as a function of time. Then he could have found the distance he travelled by integrating $g(t)$ over the time period of his trip.

Rewrite and then restate the expressions above that involve $g$ in terms of the function $f$ instead. Everywhere you see "$g(t)$", just change it to "f'(t)".

First fundamental theorem

Rearranging this just a little bit: $$F(b)=F(a)+\color{blue}{\int_a^b F'(x)\,dx}.$$ Which we can interpret as:

$F(b)$, the value of the function at $b$,
    is equal to
$F(a)$, the value of the function at $a$,
    plus
the accumulated change of $F$, from $a$ to $b$.

I will sometimes write $F(b)-F(a)\equiv \Delta F$.

Second fundamental theorem