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Chapter 3 1 - 3 Cl DIFFERENTIATION RULES 3.1 Derivatives of...

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Unformatted text preview: 3 Cl DIFFERENTIATION RULES 3.1 Derivatives of Polynomials and Exponential Functions eh—l 1. (a) e is the number such that lim 11—0 ('3) 2 7h — 1 . 2.8h — 1 , From the tables (to two decimal places) [111% h : 0.99 and $1111) : 1.03. Since 0.99 < 1 < 1.03, 2.7 < e < 2.8. 2. (a) (b) f(;v) : e“ is an exponential function and g(w) : we is a - a: d e _ e—l power function. E (e ) — er and dar (:c ) 7 ea: . (c) f ($) : e“: grows more rapidly than 9(1) 2 are when so is large. The function value at a: 2 0 is 1 and the slope at :c : 0 is 1. 3.f f(:c) : 186.5 is a constant function so its derivative is 0 that is. f'(m )— — 0. 4.f f:(ac) \/3—0 18 a constant function so its derivative IS 0 that IS f (1:): 0. 5.f(:c):593~1=> f(w)=5~0:5 6.F F:(m) —4$10 :> F013): —4(10m10— 1) : 410359 7f f(:c)=:c2 +3m—4 => f’(m)=2m2_1+3— 022m+3 8. g(:I:) — 52:8 —2w5 +6 => 9 (m) — 5(8$8_1) ~2(5m5 ) +0 =40$7 ~10$4 9.x f—(t) — W + 8) 2» f (t) = at“ + 8) =1<4t4‘1+ 0) = t3 10.f f(t) 2 5t6 — 314 + t => f’(t) : %(6t5) — 3(4253) +1 : 3:5 1 1213 +1 -2/5 :> y' : —3x(_2/5)_1 : ¥gm—7/5 _ _ 2 5 11. y = m 5 _ 5m7/5 12. 31:561-1-3 :> y':5(e w)+0=5€$ 13. V(r) : gnrs :> V'(r)= 377(31"2 )— ~ 47rr2 14. R(t) : 525*“ :> R'(t) : 5[—§t(’3/5)‘1] : —3t‘8/5 151 ...
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