he Secant
line PO
jpo 全
slope of tangent line GZ+ 全 bs
二Instantaneous velocity i i
$ which is the same as the average velocity over the time interval [lg,d 十 四. Therefore
7 e
the velocity at time # 三 d (the limit of these average velocities as娘 approaches 0) must
7 [be equal to the slope of the tangent line at (the limit of the slopes of the secant lines).
Examples 1 and 2 show that in order to Solve tangent and velocity problems we must
be able to find limits. After Studying methods for computing limits in the next four sec-
tions, we Will return to the problems of finding tangents and velocities in Chapter 2
FIGURE 4
kk 1.4 EXERCISES ,
1. ,A tank holds 1000 liters of water, which drains from the (a) IPis the point (15, 250) on the graph of V, find the
bottom of the tank in half an hour. The values in the table slopes of the secant lines PO when O is the point on the
show the volume / of water in the tank (in liters) graph_ with 二5, 10, 20,25,and 30.
(b) .Estimate the slope ofthe tangent line 午 吃by averaging
after 7 minutes-
the slopes of two secant lines.
| fr(min) S$ 2 30 (c) Use agraph of the function to estimate the slope of the
tangent line at P. (This slope Tepresents the rate at which
| YOD 28 必 the Water is 1s siowing, from the tank after 15 minutes.)
