Calculus: Analytic Geometry and Calculus, with Vectors

(lu) #1
3.4 Continuity 149

is continuous. Prove the statement by filling in the intermediate steps in the
formula
lim P(x) P(a)
a-4a
and tell which theorems on limits are used in the process. Remark: The same
procedure shows that each polynomial is continuous.
2 Letting
Q(x) _


(x - 1) (x - 3)
(x - 2) (x - 4)'
show that Q is continuous at each x except 2 and 4.
3 Prove that the quotient of two functions is continuous wherever both
functions are continuous and the denominator is not zero. Remark: We recall
that the quotient of two polynomials is sometimes called a rational function.
Our results show that a rational function is continuous wherever the denominator
is not zero.
4 Determine the points of discontinuity of the functions f,, etcetera, for
which

(a) fi(x)= 1 + x2 (b) f2(x)^1

x
x2

(c) fa(x) = x(1^1 x) (d) f4(x) jxj
1 1
(e) fs(x) =x2 + 2x - 3

5 Does the assertion

(f) fn(x) =x2 + 2x + 3

approx f (x) =f(a)
..I=-al<s
abbreviate the assertion that to each positive number a there corresponds a posi-
tive number 3 such that lf(x) - f(a)i < e whenever Ix - al < 5? Ins.: It
can, but it does only if we agree that it does.
6 Taxi fare is 50 cents plus 10 cents for each quarter mile or fraction
thereof. Letting f(x) denote the fare for a ride of x miles, sketch a graph of f
and tell where f is discontinuous.
7 Assume (as is not quite true) that it takes 0.5 calorie of^220
heat to raise the temperature of 1 gram of ice 1 degree centigrade, u0
that it takes 80 calories to melt the ice at 0°C, and that it takes e0-
1.0 calorie to raise the temperature of I gram of water one degree so-
centigrade. Supposing that -40 S x < 20, let Q(x) be the +0-
number of calories of heat required to raise one gram of H2O
from temperature -40°C to x°C. Sketch a graph of Q. Ans.: ,O _20TITu
Figure 3.491.
8 The magnitude of the gravitational force which the earth Figure 3.491
exerts upon a particle is called its weight W. Suppose (as would
be true in the mechanics of Newton if the earth were a homogeneous spherical
ball) that there exist constants kl and k2 such that
W=klx (05x<R)

W=xs (xzR),
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