Difference Equations to Differential Equations (1).pdf

¢¡£¡¥¤§¦¨¤§©
¤
 £© Section

 ¢¡£¡¥¤§¦¨¤§© £
 £© Calculus: Areas And Tangents
The study of calculus begins with questions about change. What happens to the velocity of
a swinging pendulum as its position changes? What happens to the position of a as
time changes? What happens to a population of owls as its rate of reproduction changes?
Mathematically, one is interested in learning to what extent changes in one quantity affect
the value of another related quantity. Through the study of the way in which quantities
change we are able to understand more deeply the relationships between the quantities
themselves. For example, changing the angle of elevation of a projectile affects the distance
it will travel; by considering the effect of a change in angle on distance, we are able to
determine, for example, the angle which will maximize the distance.
Related to questions of change are problems of approximation. If we desire to approxi-
mate a quantity which cannot puted directly (for example, the area of some planar
region), we may develop a technique for approximating its value. The accuracy of our tech-
nique will depend on how putations we are willing to make; calculus may then
be used to answer questions about the relationship between the accuracy of the approxi-
mation and the number of calculations used. If we double t