. . . . SecƟon 2.4 The Product and QuoƟent Rules V63.0121.001: Calculus I Professor MaƩhew Leingang New York University February 23, 2011 . Announcements I Quiz 2 next week on §§1.5, 1.6, 2.1, 2.2 I Midterm March 7 on all secƟons in class (covers all secƟons up to 2.5) . Help! Free resources: I Math Tutoring Center (CIWW 524) I College Learning Center (schedule on Blackboard) I TAs’ office hours I my office hours I each other! . Notes . Notes . Notes . 1 . . SecƟon 2.4: Product/QuoƟent Rule . V63.0121.001: Calculus I . February 23, 2011
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Lesson 9: The Product and Quotient Rules (handout)
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Sec on 2.4The Product and Quo ent Rules
V63.0121.001: Calculus IProfessor Ma hew Leingang
New York University
February 23, 2011
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Announcements
I Quiz 2 next week on§§1.5, 1.6, 2.1, 2.2
I Midterm March 7 on allsec ons in class (coversall sec ons up to 2.5)
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Help!Free resources:
I Math Tutoring Center(CIWW 524)
I College Learning Center(schedule on Blackboard)
I TAs’ office hoursI my office hoursI each other!
.
Notes
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Notes
.
Notes
. 1.
. Sec on 2.4: Product/Quo ent Rule. V63.0121.001: Calculus I . February 23, 2011
I Understand and be ableto use the Product Rulefor the deriva ve of theproduct of two func ons.
I Understand and be ableto use the Quo ent Rulefor the deriva ve of thequo ent of twofunc ons.
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OutlineDeriva ve of a Product
Deriva onExamples
The Quo ent RuleDeriva onExamples
More deriva ves of trigonometric func onsDeriva ve of Tangent and CotangentDeriva ve of Secant and Cosecant
More on the Power RulePower Rule for Nega ve Integers
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Recollection and extension
We have shown that if u and v are func ons, that
(u+ v)′ = u′ + v′
(u− v)′ = u′ − v′
What about uv?
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Notes
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Notes
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Notes
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. Sec on 2.4: Product/Quo ent Rule. V63.0121.001: Calculus I . February 23, 2011
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Is the derivative of a product theproduct of the derivatives?
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.(uv)′ = u′v′?
.(uv)′ = u′v′!
Try this with u = x and v = x2.I Then uv = x3 =⇒ (uv)′ = 3x2.I But u′v′ = 1 · 2x = 2x.
So we have to be more careful.
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Mmm...burgersSay you work in a fast-food joint. You want to make more money.What are your choices?
I Work longer hours.I Get a raise.
Say you get a 25 cent raise inyour hourly wages and work 5hours more per week. Howmuch extra money do youmake?
...∆I = 5× $0.25 = $1.25?
.∆I = 5× $0.25 = $1.25?.∆I = 5× $0.25 = $1.25?
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Money money money moneyThe answer depends on how much you work already and yourcurrent wage. Suppose you work h hours and are paid w. You get ame increase of∆h and a wage increase of∆w. Income is wagesmes hours, so
∆I = (w+∆w)(h+∆h)− whFOIL= w · h+ w ·∆h+∆w · h+∆w ·∆h− wh= w ·∆h+∆w · h+∆w ·∆h
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Notes
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Notes
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Notes
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. Sec on 2.4: Product/Quo ent Rule. V63.0121.001: Calculus I . February 23, 2011
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A geometric argumentDraw a box:
..w
.∆w
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h
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∆h
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wh
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w∆h
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∆wh
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∆w∆h
∆I = w∆h+ h∆w+∆w∆h
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Cash flowSupose wages and hours are changing con nuously over me. Overa me interval∆t, what is the average rate of change of income?
∆I∆t
=w∆h+ h∆w+∆w∆h
∆t
= w∆h∆t
+ h∆w∆t
+∆w∆h∆t
What is the instantaneous rate of change of income?
dIdt
= lim∆t→0
∆I∆t
= wdhdt
+ hdwdt
+ 0
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Eurekamen!We have discoveredTheorem (The Product Rule)
Let u and v be differen able at x. Then
(uv)′(x) = u(x)v′(x) + u′(x)v(x)
in Leibniz nota on
ddx
(uv) =dudx
· v+ udvdx
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Notes
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Notes
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Notes
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. Sec on 2.4: Product/Quo ent Rule. V63.0121.001: Calculus I . February 23, 2011