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Chapter 3 18

# Chapter 3 18 - 168 CHAPTER 3 DIFFERENTIATION RULES ﬂ ﬂ...

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Unformatted text preview: 168 CHAPTER 3 DIFFERENTIATION RULES ﬂ_ ﬂ dm_ dx the rate at which the volume is increasing as ac increases past 3 mm. 12. (a) V(:c) = m3 => 3302. = 3(3)2 : 27 mm3/mm is 1:3 (b) The surface area is 3(33) 2 6:132. so V/(x) = 31:2 = %(6m2) 2 £302). The ﬁgure suggests that if A3: is small. then the change in the volume of the cube is approximately half of its surface area (the area of 3 of the 6 faces) times Aw. From the ﬁgure. AV 2 3332(Am) —I— 337(A\$)2 + (Am)? If A2: is small. then AV % 3m2(Am) and so AV/Am m 3m2. 13. (a) Using A0") : 7W2. we ﬁnd that the average rate of change is: . 14(3) — 14(2) _ 971' 7 47r _ .. A(2.5) * 14(2) 7 6.257r i 47r _ (1) ————3 _ 2 — ———1 — 57r (n) 2.5 _ 2 — 0'5 — 4.5a A(2.1) 7 A(2) 7 4.417r , 47r g (1“) 2.1 — 2 — 041 — 4‘1” (b) A(r) 2 71'7'2 :> A'(7“) = 271'7”, so A’(2) = 47r. (c) The circumference is C (r) = 2777’ : A' (r) The ﬁgure suggests that if AT is small. then the change in the area of the circle (a ring around the outside) is approximately equal to its circumference times Ar. Straightening out this ring gives us a shape that is approximately rectangular with length 2m" and width Ar. so AA m 27rr(Ar). Algebraically. AA : A(r + A7") — A(r) = 7r(r + A1")2 — 7W2 2 27rr(A7’) + 7r(A7‘)2. So we see that if AT is small. then AA % 27rr(Ar) and therefore. AA/Ar % 27W. 14. Aftert seconds the radius is 7' = 6015. so the area is A(t) : 7r(60t)2 : 36007rt2 :> AI (t) : 72007rt => (a) A’(1) : 7200a cmZ/s (b) A’(3) : 21.6007r ch/s (c) A’(5) : 36.0007r cm2 /s As time goes by. the area grows at an increasing rate. In fact. the rate of change is linear with respect to time. 15. 5(1") : 47rr2 :> S’(r) : 87W :> (a) S’(1) = 87r ft2 /ft (b) S'(2) : 16w ft2/ft (c) S’(3) 2 24a ft2 /ft As the radius increases. the surface area grows at an increasing rate. In fact. the rate of change is linear with respect to the radius. 16. (a) Using V(r) : gang. we ﬁnd that the average rate of change is: V(8) # V(5) : §7r(512) 7 §w(125) (i) 8 _ 5 3 : 1727f pm3/pm , 4 216 — 3 125 _ (ii) M : 37“ I 37“ I 2 121.371’ pma/[Lm 6 — 5 1 :1 3 _ g 3 _ (iii) W : 37r(5.1)0 1 37r(5) : 102.0137r pms/pm (b) V’(r) : 47rr2. so V’(5) : 10071' hm3/pm. ...
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