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d. f(0) = = lim x-+0 = lim x-t 0 rim !d:{Q h+0 x-u x-sin2x sln x (-1) 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 , . 1.1666... x 10-9 _ ^ ^^r {a Quotient = = 0.001 16... . 9.999... x 10-7 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 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 3x4 + 12x3h + 18x2h2 + 12xh3 + 3h4 - 3x4 h-+0 = lim (12x3+ T . t(x) = cos 3x + t'(x) = -3 sin 3x. t'(5) = -3 sin 15 = -1.95086... Decreasing at 1.95... y-units per x-unit. 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. T6. Amos substituted before differentiating instead ol after. Correct solution is f(x) = 7x + f'(x) =7 + f'(5) = 7. h 18x2h + 12xh2 + 3h3; = 12xs, Q.E.D. T7. Graph. f(x) = i!!I As x approaches 0, f(x) approaches 1. The squeeze theorem. lt states that: lf (1) g(x) < h(x) for all x in a neighborhood of c, (2) ,lg s(x) = Jg h(x) = l-, 6n6 (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. *q{sin 5x) = 5 cos 5x T1 1.y=60x23-x +25 + y'=4Oy1t3-1 T12. f(x) = cos (sin5 7x) + f'(x) = -si6 (sins zx) . 5 sina 7x. cosTx .7 = -35 sin (sins 7x) sinaTx cos 7x T13. y'= 0.6 . . (Function is y = -3 + 1.5x, for which the numerical derivative is 0.6081... .) 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 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

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h(x) = l-, 6n6 (3) f is a function for which g(x) < f(x) < h(x) for all x in that the graph appears to get closer and closer to the tangent line. The name of this property is local linearity. f'(x) = -si6 (sins zx) . 5 sina 7x. cosTx .7 = lim (12x3+ T8. f(x) = (7x + 3)1s - f'(x) = 105(7x + 3)14 T9. g(x) = cos (x5) + g'(x) = *5x4 sln xs (-1) T13. y'= 0.6 . . i!!I 18x2h + 12xh2 + 3h3; = 12xs, Q.E.D. 9.999... x 10-7 T3. Prove that if f(x) = 3t', then f'(x) = llx3. d = V'= 20.8f3'6 sln x 44

TRANSCRIPT

d. f(0) =

= limx-+0

= limx-t 0

rim !d:{Qh+0 x-u

x-sin2xsln x (-1)

xx-sin2x+sinx

xstnxUsing TABLE for numerator, denominator, andquotient shows that the numerator goes to zerofaster than the denominator. For instance, ifx = 0.00'l ,

. 1.1666... x 10-9 _ ^ ^^r {aQuotient = = 0.001 16... .

9.999... x 10-7Thus, the limit appears to be zero.(The limit can be found algebraically to equal zero byI'Hospital's rule after students have studied Section6-8.)

Chopter TestT1. 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

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

h-+0

= lim (12x3+

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

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

T6. Amos substituted before differentiating instead olafter.Correct solution is f(x) = 7x + f'(x) =7 + f'(5) = 7.

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

T7. Graph. f(x) = i!!I

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

lf (1) g(x) < h(x) for all x in aneighborhood of c,

(2) ,lg s(x) = Jg

h(x) = l-, 6n6

(3) f is a function for whichg(x) < f(x) < h(x) for all x in thatneighborhood 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. *q{sin 5x) = 5 cos 5x