2602.sagews

Discovering the Fundamental Theorem of Calculus

The Section 5.3 concept review opens with a graph of velocity versus time for a bicycle trip.  After answering the questions on that slide, we now construct a table of values of the distance function $s(t)$.  These values are found by calculating area bounded by the graph of velocity. 

c=0
t=var('t')
s0=c
s1=c+.5*1*30*60
s2=s1+1*30*60
s3=s2+.5*1*30*60
s4=s3
s5=s4
s6=s5-.5*1*30*60
s7=s6-1*30*60
s8=s7-1*30*60
s9=s8-.5*1*30*60
s10=s9
s11=s10+.5*1*30*60
splot=point([(0,s0), (1,s1), (2,s2), (3,s3), (4,s4), (5,s5),(6,s6), (7,s7), (8,s8), (9,s9), (10,s10), (11,s11)])
show(splot)

What are we using here?  $$s(b)=s(a) + \int^b_a v(t) \, dt.$$

This is true because velocity is the rate of change of position.  Stated more generally, this is the

Fundamental Theorem of Calculus (FTC):  If $f$ is continuaous on the interval $[a,b]$ and $f(t)=F'(t)$, then

$$\int^b_a f(t) \, dt = F(b)-F(a).$$

 

Notes: 

1.  This theorem establishes the relationship between the two operations we developed in this course: finding derivatives (slopes, rates of change) and finding integrals (areas, cummulative change).

2.  Before computers, this theorem provided a practical way of calculating integrals in many cases.

Example:  Find the area bounded by the curves $y=3x^2$, $y=0$, and $x=3.$ 

boundary1=plot(3*x^2,(0,3.2))
boundary2=line([(0,0),(3,0)])
boundary3=line([(3,0),(3,27)])
show(boundary1+boundary2+boundary3)

This bounded area is $$\int^3_0 x^2 \, dx.$$

Can you think of a function $F(x)$ such that $F'(x) =3 x^2?$ 

If so, then FTC says $$\int^3_0 3x^2 \, dx = F(3)-F(0).$$

Recall the definition of the integral as a limit of Riemann sums: $$\int^b_a f(t) \, dt = \lim_{n\rightarrow \infty} \sum^n_{i=1} f(t_i) \Delta t.$$

Explain each symbol in the sum.

Substituting $\frac{b-a}{n}$ for $\Delta t$ yields $$\int^b_a f(t)\, dt = \lim_{n\rightarrow \infty} \frac{b-a}{n}\sum^n_{i=1}f(t_i).$$

Dividing both sides by $b-a$ yields $$\frac{1}{b-a} \int^b_a f(t) \, dt = \lim_{n\rightarrow \infty} \frac{1}{n}\sum^n_{i=1}f(t_i).$$

If we ignore the limit, the expression on the right-hand side looks like an average. 

Explain what is being "averaged."

What happens as $n$ increases without bound?

The expression on the left-hand side is defined to be the average value of $f$ on the interval $[a,b]$.  Now finish the Section 5.3 concepts review.

Extra Question:  In the bicycle scenario, what happens if we start at position $x(0)=500$ instead of $x(0)=0$? 

c=500
t=var('t')
s0=c
s1=c+.5*1*30*60
s2=s1+1*30*60
s3=s2+.5*1*30*60
s4=s3
s5=s4
s6=s5-.5*1*30*60
s7=s6-1*30*60
s8=s7-1*30*60
s9=s8-.5*1*30*60
s10=s9
s11=s10+.5*1*30*60
splot=point([(0,s0), (1,s1), (2,s2), (3,s3), (4,s4), (5,s5),(6,s6), (7,s7), (8,s8), (9,s9), (10,s10), (11,s11)])
show(splot)