d. f(0) = rim

!d:{Q

T7. Graph. f(x) =

h+0 x-u

x-sin2x

lim = x-+0 lim = x-t 0

i!!I

(-1)

sln x x

x-sin2x+sinx xstnx

Using TABLE for numerator, denominator, and quotient shows that the numerator goes to zero faster than the denominator. For instance, if x = 0.00'l

As x approaches 0, f(x) approaches 1. The squeeze theorem. lt states that:

,

. Quotient

1.1666... x 10-9 _ {a = 0.001 16... . = 9.999... x 10-7 ^ ^^r Thus, the limit appears to be zero. (The limit can be found algebraically to equal zero by I'Hospital's rule after students have studied Section

lf (1) g(x) < h(x) for all x in a neighborhood of c, (2) h(x) = l-, 6n6 s(x) =

6-8.)

Chopter Test T1. See definition of limit in Chapter 2. T2. See definition of derivative in Section 3-2 or 3'4. T3. Prove that if f(x) = 3t', then f'(x) = llx3. Proof

:

*;,l.a"tire=,1'lrg# lim = h-+0

= T

3x4 + 12x3h + 18x2h2 + 12xh3 + 3h4

-

3x4

h

lim (12x3+ 18x2h + 12xh2 + 3h3; = 12xs, Q.E.D.

. t(x) = cos

3x

+

t'(x) =

Jg

,lg

(3) f is a function for which g(x) < f(x) < h(x) for all x in that neighborhood of c,

Then lim f(x) = L. T8. f(x) = (7x + 3)1s - f'(x) = 105(7x + 3)14 T9. g(x) = cos (x5) + g'(x) = *5x4 sln xs

tto. T1

5x) = 5 cos 5x

*q{sin

1.y=60x23-x +25

+

y'=4Oy1t3-1

T12. f(x) = cos (sin5 7x) + f'(x) = -si6 (sins zx) . 5 sina 7x. cosTx = -35 sin (sins 7x) sinaTx cos 7x

.7

T13. y'= 0.6 . . (Function is y = -3 + 1.5x, for which the numerical

-3 sin 3x.

t'(5) = -3 sin 15 = -1.95086... Decreasing at 1.95... y-units per x-unit.

derivative is 0.6081... .)

T5. Graph (below). lf you zoom in on the point where x = 5, the graph appears to get closer and closer to the tangent line. The name of this property is local

linearity.

T14.v=3+5rl

6

v = Y'= -8x-2'6 d = V'= 20.8f3'6 T15. f'(x) -72vsta + f(x) =gzxs4 T16. f'(x) = 5 sin x and f(0) = 13

f(x)=-5cosx+C 13=-5cos0+C =+ C= 18 f(x)=-5cosx+18

T17. Carbon Dioxide Problem D'

c(t)=300+Zcosffit T6. Amos substituted before differentiating instead ol after. Correct solution is f(x) = 7x

+

f'(x)

=7 +

f'(5) = 7.

a. c'(t)

=-ffi"i"fr,

b. c'(273)

=-#sin

1ft.zzs1

= 0.03442... ppm/day c. Rate

"#ffi, " m=

23e0.66...,

which is approximately 2390 tons per second!

44

Colculus: Concepts ond Applicotions

Problem Set 3-10

IMG_0018

h(x) = l-, 6n6 (3) f is a function for which g(x) &lt; f(x) &lt; h(x) for all x in that the graph appears to get closer and closer to the ta...