Section 2.4 The Product and Quotient Rules V63.0121.041, Calculus I New York University October 3, 2010 Announcements I Quiz 2 next week on §§1.5, 1.6, 2.1, 2.2 I Midterm in class (covers all sections up to 2.5) Announcements I Quiz 2 next week on §§1.5, 1.6, 2.1, 2.2 I Midterm in class (covers all sections up to 2.5) V63.0121.041, Calculus I (NYU) Section 2.4 The Product and Quotient Rules October 3, 2010 2 / 40 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! V63.0121.041, Calculus I (NYU) Section 2.4 The Product and Quotient Rules October 3, 2010 3 / 40 Notes Notes Notes 1 Section 2.4 : The Product and Quotient Rules V63.0121.041, Calculus I October 3, 2010
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Section 2.4The Product and Quotient Rules
V63.0121.041, Calculus I
New York University
October 3, 2010
Announcements
I Quiz 2 next week on §§1.5, 1.6, 2.1, 2.2
I Midterm in class (covers all sections up to 2.5)
Announcements
I Quiz 2 next week on §§1.5,1.6, 2.1, 2.2
I Midterm in class (covers allsections up to 2.5)
V63.0121.041, Calculus I (NYU) Section 2.4 The Product and Quotient Rules October 3, 2010 2 / 40
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!
V63.0121.041, Calculus I (NYU) Section 2.4 The Product and Quotient Rules October 3, 2010 3 / 40
Notes
Notes
Notes
1
Section 2.4 : The Product and Quotient RulesV63.0121.041, Calculus I October 3, 2010
V63.0121.041, Calculus I (NYU) Section 2.4 The Product and Quotient Rules October 3, 2010 8 / 40
Money money money money
The answer depends on how much you work already and your currentwage. Suppose you work h hours and are paid w . You get a time increaseof ∆h and a wage increase of ∆w . Income is wages times hours, so
∆I = (w + ∆w)(h + ∆h)− wh
FOIL= w · h + w ·∆h + ∆w · h + ∆w ·∆h − wh
= w ·∆h + ∆w · h + ∆w ·∆h
V63.0121.041, Calculus I (NYU) Section 2.4 The Product and Quotient Rules October 3, 2010 9 / 40
Notes
Notes
Notes
3
Section 2.4 : The Product and Quotient RulesV63.0121.041, Calculus I October 3, 2010
A geometric argument
Draw a box:
w ∆w
h
∆h
w h
w ∆h
∆w h
∆w ∆h
∆I = w ∆h + h ∆w + ∆w ∆h
V63.0121.041, Calculus I (NYU) Section 2.4 The Product and Quotient Rules October 3, 2010 10 / 40
Supose wages and hours are changing continuously over time. Over a timeinterval ∆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?
dI
dt= lim
∆t→0
∆I
∆t= w
dh
dt+ h
dw
dt+ 0
V63.0121.041, Calculus I (NYU) Section 2.4 The Product and Quotient Rules October 3, 2010 11 / 40
Eurekamen!
We have discovered
Theorem (The Product Rule)
Let u and v be differentiable at x. Then
(uv)′(x) = u(x)v ′(x) + u′(x)v(x)
in Leibniz notationd
dx(uv) =
du
dx· v + u
dv
dx
V63.0121.041, Calculus I (NYU) Section 2.4 The Product and Quotient Rules October 3, 2010 12 / 40
Notes
Notes
Notes
4
Section 2.4 : The Product and Quotient RulesV63.0121.041, Calculus I October 3, 2010