Calculus: Analytic Geometry and Calculus, with Vectors

200 Functions, limits, derivatives

where g is the (scalar) acceleration of gravity, vo is the initial speed, and a is the
angle of elevation of the gun so 0 < a < 7r/2. Find a formula in which wecan
put estimates of errors in g, v0, and a to obtain an estimate of the resulting error
in R. Hint: Use logarithms. t1ns.:

dv0

V0 +Idgg

2a cos 2a

sin 2a

do!

Remark: When a is not too close to it/2, the factor multiplying Ida/al has the

order of magnitude of Icos 2al. When a is near 7r/2, the factor is very large.

We are now able to enlighten rabbit hunters when they ask us why they are

unlikely to hit their targets when they shoot almost straight up.

12 Find the maximum possible percentage of error in the computed estimate

of the volume of a cone that can be caused by errors not exceeding p per cent and

q per cent in measurements of the height and the base radius of the cone. Ans.:

p+2q.

13 When three resistors having resistances rr, r2, r3 are connected in parallel,

the resulting resistance R is determined by the formula

1 1 1 1

-T1+r2 .+73

With the aid of the fact that resistances are always positive, prove that if no

error in a resistor exceeds p per cent, then the error in R produced by these errors

cannot exceed p per cent. Remark: This conclusion really means something to

those who design the mazes hidden in our television sets. Problems involving

silver bands and gold bands and tolerances (percentages of error) cannot be

ignored. Engineers do not like to behave like rabbit hunters who shoot almost

straight up.

14 Show that the conclusion of the preceding problem is violently false if the

numbers rl, r2, r3 are not resistances of resistors but are numbers of which some

can be positive and some can be negative.

15 Let r be a differentiable function of t. For example, we may have

(1) r(t) = x(t)i + y(t)j + z(t)k,

where all of the functions are differentiable and the vectors i, j, k are the unit

orthonormal vectors of Section 2.2. By a definition analogous to the one involv-

ing scalar differentials, a vector dr and a scalar dt constitute a pair of differentials

if

(2) dr = r'(t) dt.

With or without using the rectangular representation (1) and the fact that

Ir(t)12 = [x(t)]2 + [y(t)]2 + [Z(t)]2,

prove that

dlr(t)12 =

16 The specific heat of a substance is sometimes said to be the number of

calories of heat required to raise the temperature of 1 gram of the stuff 1 degree

centigrade. This definition is sometimes useful but, because substances have

different specific heats at different temperatures, the following definition is much