Calculus: Analytic Geometry and Calculus, with Vectors

198 Functions, limits, derivatives

4

(1)

Suppose that x and y are differentiable functions of t such that

x2+y2 = 1.

Show that differentiating with respect tot and multiplying by dt gives the formula

(2) x dx + y dy =0.

Remark: In case t = x, we can divide (2) by dx and recover the first of the formulas

(3) x + ydx= 0, xdy+ Y = 0,

which is valid when y is a differentiable function of x for which (1) holds. In

case t = y, we can divide (2) by dy and recover the second formula in (3), which

is valid when x is a differentiable function of y for which (1) holds.

5 Supposing that n is a constant and x is positive, observe that the first

of the formulas

(1) y = x", log y = n log x

is equivalent to the second. Use these formulas to obtain

(2)

and

(3)

dy = nx' dx

1dy=n1dx.

Use (2) to show that if ldxl 5 (p/100)x, then idyl 5 (Inlp/100)y. Then use (3)

to obtain the same result.

6 Gradual changes in tensions or compressions or temperatures can produce

gradual changes in the lengths of iron rods that form a triangle such as that

u

Figure 3.991

shown in Figure 3.991. Therefore, because the law of cosines

must hold at each time t, it makes sense to suppose that we

have four positive differentiable functions oft such that

(1) w2 = u2 + v2 - 2uv cos 0.

Equate the derivatives with respect to t of the members of (1)

and multiply by dt to obtain the differential formula

(2) w dw = (u - v cos 6) du + (v - u cos 0) dv + uv sin 0 d8.

Remark: It is sometimes both possible and unwise to underestimate potentialities

of formulas. The formulas (1) and (2) contain eight numbers u, v, w, 0, du, dv,

dw, and dB. There are many situations in which some information about some

of these numbers is known and the two formulas can be used to eke out more

information. Some problems are much more recondite than the one solved by

finding w from (1) and then finding dw from (2) when the values of the other six

numbers are known.

(^7) Supposing that T, L, and g are positive, observe that the first of the
formulas
(1) T = 2a Zig, L 1 1
71, = 41r2 g, log T= log2a -
2 log L - 2 log g