# unit 4 calculus and vectos.pdf - 1a lim u2212 2 3 u21920...

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1a) ? 0 lim 2?𝑖? 3? =− 25𝑖?(3. 0) = 0 b) ?𝑖? 4??? 2? = 4 ?𝑖? ???2? = 4[???(0))] = 4[???(0)] = 4(1) = 4 2a) ? = ?𝑖?? 5 ?? ?? = ? ?? (?𝑖?(? 5 )) = ???(? 5 ) ? ?? (? 5 ) ?? ?? = 5? 4 ???? 5 b) ? = ?𝑖? 5 ? ?? ?? = ? ?? (?𝑖? 5 ?) = 5?𝑖? 4 ? ? ?? (?𝑖??) ?? ?? = 5?𝑖? 4 ? ???? c) ? = ???(4? + 4) ?? ?? = ? ?? [???(4? + 4)] =− ?𝑖?(4? + 4) ? ?? (4? + 9) ?? ?? = ? ?? [??? 2 (? 2 + 3)] d) ? = ??? 2 (? 2 + 3) ?? ?? = ? ?? ??? 2 (? 2 + 3) [ ] = 2???(? 2 + 3) · ? ?? [???(? 2 + 3)] = 2???(? 2 + 3) − ?𝑖?(? 2 + 3) · ? ?? (? 2 + 3) = ????(? 2 + 3) − ?𝑖?(? 2 + 3) − 2? [ ] = ?? ?? =− 4????(? 2 + 3)?𝑖?(? 2 + 3) e) ? = ? 2 ?𝑖?(? 2 + 5) ?? ?? = ? ?? ? 2 ?𝑖?(? 2 + 5) [ ] = ? 2 ? ?? (?𝑖?(? 2 + 5)) + ( ? ?? (? 2 ))?𝑖?(? 2 + 5) = ? 2 ???(? 2 + 5) ? ?? (? + 5) + 2? ?𝑖?(? 2 + 5)
= ? 2 ???(? 2 + 5)(2?) + 2? ?𝑖?(? 2 + 5) ?? ?? = 2? 3 ???(? 2 + 5) + 2? ?𝑖?(? 2 + 5) f) ? = 5? ? ?? ?? = ? ?? (5? ? ) = 5 · ? ?? (? ? ) = 5? ? ?? ?? = 5? ? g) ? = ? 5 ? ?? ?? = ? ?? (? 5? ) = 5? 5? ?? ?? = 5? 5? h) ? = ?? 3?+1 ?? ?? = ? ?? (?? 3?+1 ) = ? 3?+1 · ? ?? (?) + ? ? ?? (? 3?+1 ) = ? 3?+1 + ?? 3?+1 · ? ?? (3? + 1) = ? 3?+1 + 3?? 3?+1 ?? ?? = ? 3?+1 (3? + 1) i) ? = ??(? + ?) ?? ?? = ? ?? ??(? + ?) = 1 ?+3 ? ?? (? + 3) ?? ?? = 1 ?+3 j) ? = 5??(? 2 + 4) ?? ?? = ? ?? 5??(? 2 + 4) = 5 ? 2 +4 · ? ?? (? 2 + 3) = 5 ? 2 +4 · (2?) ?? ?? = 10? ? 2 k) ? = 5 ?
?? ?? = ? ?? 5 ? ?? ?? = 5 ? ??5 L) ? = (? 2 + 2) 2 (? 2 + 2? + 1) 3 ? = (? 2 + 2) 2 (? + 1) 2 ( ) 3 ? = (? 2 + 2) 2 (? + 1) 6 ?? ?? = ? ?? (? 2 + 2) 2 (? + 1) 6 [ ] =(? + 1) 6 ? ?? (? 2 + 2) 2 + (? 2 + 2) 2 · ? ?? (? + 1) 6 =(? + 1) 6 2(? 2 + 2) · ? ?? (? 2 + 2) + (? 2 + 2) 6(? + 1) 5 · ? ?? (? + 1) 5 = (? + 1) 6 · 2(? 2 + 2)(2?) + 6(? 2 + 2)(? +) 5 = 4?(? + 1) 6 (? + 2) + 6(? 2 + 2) 2 (? + 5) 5 = (? + 1) 5 (? 2 + 2) 4?(? + 1) + (6(? 2 + 21)) [ ] = (? + 1) 5 (? 2 + 2) 4? 2 + 4? + 6? 2 + 12 [ ] ?? ?? = (? + 1) 5 (? 2 + 2) 10? 2 + 4? + 12 [ ] m) = ? = ??(?𝑖??) ?? ?? = ? ?? ?? (?𝑖??) = 1 ?𝑖?? · ? ?? ?𝑖?? = ???? ?𝑖?? ?? ?? = ??? ? n) ? = (? + ???) 5 ?? ?? = ? ?? (? + ???) 5 = 5(? + ???) 4 · ? ?? (? + ???) = 5(? + ??) 4 (1 + 1 ? ) ?? ?? = 5 ? (? + 1)(? + ???) 4 3) ? = (? + 3) 2 (? + 2) 5 , (? + 4) 7 Multiply Log both sides
log(y) = ???[(? + 3) 2 ] (? + 3) 5 · (? + 4) 7 [ ] = ???(?) = ???(? + 3) 2 − ??? (? + 2) 5 · (? + 4) 7 [ ] ??? ? ? = ??? ? ??? ? ???(?. ?) = ??? ? + ??? ? [ ] ???(?) = 2 ???(? + 3) − ???(? + 3) 5 + ???(? + 4) 7 [ ] ???(?) = ???(? + 3) − 5???(? + 2) − 7???(? + 4) Differentiating both sides with respect to '?' ? ?? (???(?)) = 2 ? ?? ??? (? + 3) − 5 ? ?? (???(? + 2)) − 7 ? ?? (???(? + 4)) 1 ? ?? ?? = 2? 1 (?+3) − 5? 1 (?+2) 7 (?+4) ?? ?? = ? 2 (?+3) 5 (?+3) 7 (?+4) ?? ?? = ? 2 ?+3 5 ?+2 7 ?+4 = ? 2(?+2)(?+4) (?+3)(?+2) −5(?+3)(?+3) (?+4) −7(?+3)(?+2) = ? −10? 2 −58?−86 ?+3 ( )(?+2)(?+4) = (? + 3) 2 −10? 2 (?+3) −58? (?+2) −86 (?+4) (? + 2) 5 · (? + 4) 7 = ?? ?? = (?+3) (?+2) 6 ·(10? 2 −58?−86) ·(?+4) 4.) when x=3 ? = ??(? 3 − 2) At x=3 ? = ??(3 2 − 2) ? = ??(4 − 2) ? = ?? 7 Now state ‘m’ ⇒ ? = ?? ?? ( ) ? = 3 ? = ? ?? (??(? 2 − 2) = ? ?(? 2 −2) ??(? 2 − 2))? ? ?? (? 2 − 2) = 1 (? 2 −2) ? 2? ? = 2? ? 2 −2 ?? ? = 3, ? = 6 7 General at line at point and slope = m ??? ? 1 , ? 2
? − ? 1 = ?(? − ? 1 ) Here ? 1 = 3, ? 1 = ?? 7, ? = 6 7 ? − ??7 = 6 7 (? − 3) 5.) ? = ??? 2 ? ?? ? = π 3 , ? = (??? π 3 ) 2 ? = 1 2 ( ) 2 ? = 1 4 Now, slope ‘m’ for the tangent line will be ? = ?? ?? = ? (??? 2 ?) ?? ? = ?(??? 2 ?) ? ???? ? ?(????) ?? ? = ? = ?(??? 2 ?) ? ???? ? ?(????) ?? = 2 ???? ? − ?𝑖?? =− 2 ?𝑖?? − ???? At ? = π 3 , ? =− 2 ?𝑖? π 3 ? ??? π 3 ( ) =− 2 ? 3 2 ? 1 2 =− 3 2 ?𝑖? π 3 = 3 2 At ? = π 3 , ? = 1 4 , ? = − 3 2 ⇒ ? − 1 4 = − 3 2 (? − π 3 ) 6.) given that: ? ? ?? ?? = ? ?? (? ? ) = ? ? = ????? ?? ??????? Slope of the line 2? ? = 7 𝑖? = −2 −1 = 2(? = − ? ? ) Since the tangent to the curve (,) is parallel to the Line 2? ? = 7, ?ℎ?? ?????? ??? ????? 𝑖, ?, ?? ?? = 2 ⇒ ? ? = 2 ⇒ ? = ??(2)
Put in 1 we have ? = ??(2) ? = ? ??(2) = 2 The point on the curve is (??(2), 2) Therefore required equation of the tangent at (𝑖?(2, 2) 𝑖? ? − 2 = 2(? − ??(2)) → ? = 2? 2 ??(2) + 2 7) ?(?) = ? 3 ? ? ?(?) = ? 3 ? ? + 3? 2 ? ? = ? 3 ? ? (? + 3) For increasing ? 1 (?) > 0 ⇒ ? 2 ? ? (? + 3) > 0 ⇒ ? + 3 > 0, ? ≠ 0 ? >− 3, ? ≠ 0 Increasing (− 3, 0) ∪ (0, ∞) Decreasing (− ∞, − 3) 8) ?(?) = ?