Introduction to the Fundamental Theorem of Calculus

1. A particle is traveling along a straight line. Its position is given by S(t) = t^2 – 3t + 1. Find the change in position from t = 1 to t = 4.

∆y/∆x = [s(4)-s(1)]/(4-1) = [(4^2-3*4+1)-(1^2-3*1+1)]/(4-1) = 2

The question is asking for the change in position, which is the slope. Since time is the independent variable and position is the dependent variable, the derivative of the graph is the velocity, which is the slope.

2. Suppose a car is moving with non-decreasing speed according to the table below:
t (sec) 0246810
speed (ft/sec) 303640485460

a) What is an upper estimate for the distance traveled in the first 2 seconds?

36*2 = 72ft

b) Determine upper and lower estimates for the change in position for the first 10 seconds.

Upper estimate = 36*2+40*2+48*2+54*2+60*2 = 476
Lower estimate = 30*2+36*2+40*2+48*2+54*2 = 416

b
∫y’dx = change in y over the interval [a,b]
a

The definition of the Fundamental Theorem of Calculus is on Slide 6.

http://apcalc07.blogspot.com/2007/10/todays-slides-october-25.html

http://apcalc07.blogspot.com/2007/10/thursdays-post.html

END NOTES:
3.4

Chpater 3 Supplementary Problems
Chapter 3 Pre-Test and Test is postponed to next week.
Next scribe is Yinan.