Calculus: Analytic Geometry and Calculus, with Vectors

3.7 Rates, velocities 187

22 Let a particle of mass m move in a vertical plane in such a way that its

coordinates x, y are differentiable functions oft (time) and the vector r(t) running

from the origin to the particle at time t is

r(t) = x(t)i + y(t)j.

The kinetic energy E, of the particle can then be calculated from the formula

EI =mIv(t)12.

Assuming that a constant gravitational force -mgj is exerted upon the body,

we can calculate its gravitational potential energy E2 from the formula

E2 = mgy(t)

In appropriate circumstances, the number E defined by

E=E1+E2

is the total energy of the particle. For present purposes we do not need basic

information about these things, but we should know enough calculus to be able

to calculate the total energy at time t of a projectile of mass m for which

r(t) = coti + (cit - .gt2)j.

Do it. r?ns.: E = 1m(co + ci). Remark: The fact that E has the same value

at all times is no surprise to persons who know about "conservation of energy."

23 Supposing that

(1) r(t) = p(cos 4i + sin 4)j),

where p and 4) are functions oft having two derivatives, differentiate to obtain

r'(t) and then differentiate again to obtain r"(t). Ans.:

(2) r (t)

d2 d`

-- P^2 (dt

(cos 4)i + sin ¢j)

2

-I- [2 ap a + pa] (-sin Oi +cos 4)j).

24 Give a full statement of reasons why it is true that if p and 4) are functions

of i such that p has one derivative and ¢ has two derivatives, then the product

p2dohas one derivative and

d 2 d_ dp Lo 2 d20

dtp dt -Zpdt dt +

P

dt2

25 We can pick up assorted ideas by thinking about income tax rates. Let

I(x) denote the income tax which T, a taxpayer in a particular class, must pay

when his net taxable income is x. Of course x and I(x) are to be measured in

appropriate units such as dollars or marks or kilobucks. A government may

decree that if xl < x S x2, then I(x) is yi plus ki per cent of the excess of x over

xi. We can simplify this by letting ml = kl/100 and writing

I(x)= y,+ mi(x - xi) (Xi S x < x2)