Calculus: Analytic Geometry and Calculus, with Vectors

3.7 Rates, velocities 181

Supposing that 0 < t < t + At, work out a formula for the average speed of the
stone over the time interval from t to t + At. Then work out a formula for the
speed at time t. 11ns.: 10 + 32t + 16 At and 10 + 32t.
2 A vertical y axis has its positive part above the origin. A particle moves
upon this axis in such a way that its coordinate at each time t is

y = -1it2+Bt+C,

where 11 is a positive number. Show that the scalar velocity v is

v = - 2 At + B,

that the particle is going up when t < B/2A, and that the particle is going down
when t > B/211. Show that the scalar acceleration is always -2A. Show that
the greatest height attained by the particle is B2/411 + C.
3 In the context of the preceding problem, so determine 11, B, and C that
the scalar acceleration is always - 32 and the particle is 3 units below the origin
and going upward with speed 8 when t = 0.
4 A particle moves along the x axis in such a way that its x coordinate at
time t is
x=2t5-5t4-2t2-2t+1.

Find its scalar velocity and scalar acceleration at time t. ins.:

10t4-200 -4t-2, 400 -60t2-4.

5 A body moves on a line in such a way that its coordinate x at time t is

3

x=3-4t2+15t+6.

Find the time interval over which the scalar velocity is negative, and find the

distance the body moves during the interval. 11ns.: $.

6 A particle moves along an x axis in such a way that, when t z 0, its

coordinate is

x = k2t2 + c2,

where k and c are positive constants. Show that its speed is always less than

k and approaches k as t becomes infinite.

7 If an oak tree in Ohio was 20 feet tall when it was 15 years old and was

36 feet tall when it was 25 years old, the average rate of change of height (meas-

ured in feet) with respect to time (measured in years) over the 10-year interval

is 1.6 feet per year. Tell why it is not reasonable to suppose that the tree grew

steadily at the rate of 1.6 feet per year for 10 years. If the tree grew from height

30 feet to height 32 feet in a calendar year from January 1 to December 31, sketch

a reasonably realistic graph which shows how the height of the tree might depend

upon t during the year.

8 The charge Q (measured in coulombs) on an electrical capacitor at time

t is Qo sin wt, where Qo and w (omega) are constants. The rate of change of Q

with respect tot (measured in coulombs per second, that is, in amperes) is the