Definition and Properties of the Exp FunctionDefinitionPropertiesAnother DefinitionDifferentiation and GraphingChain RuleGraphingIntegrationu-SubstitutionArbitrary PowersArbitrary PowersOther BasesDefinition and Properties Differentiation Integration Arbitrary PowersLecture 4Section 7.4 The Exponential FunctionSection 7.5 Arbitrary Powers; Other BasesJiwen HeDepartment of Mathematics, University of [email protected]://math.uh.edu/∼jiwenhe/Math1432Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 1 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another DefinitionNumber eDefinitionThe number e is defined byln e = 1i.e., the unique number at whichln x = 1.RemarkLet L(x) = ln x and E (x) = exfor x rational. ThenL ◦E (x) = ln ex= x ln e = x,i.e., E (x) is the inverse of L(x).Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 2 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another DefinitionNumber eDefinitionThe number e is defined byln e = 1i.e., the unique number at whichln x = 1.RemarkLet L(x) = ln x and E (x) = exfor x rational. ThenL ◦E (x) = ln ex= x ln e = x,i.e., E (x) is the inverse of L(x).Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 2 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another Definitionex: Inverse of ln xDefinitionThe exp function E (x) = exis theinverse of the log function L(x) = ln x:L ◦E (x) = ln ex= x, ∀x.Propertiesln x is the inverse of ex: ∀x > 0, E ◦ L = eln x= x.∀x > 0, y = ln x ⇔ ey= x.graph(ex) is the reflection of graph(ln x) by line y = x.range(E ) = domain(L) = (0, ∞),domain(E ) = range(L) = (−∞, ∞).limx→−∞ex= 0 ⇔ limx→0+ln x = −∞,limx→∞ex= ∞ ⇔ limx→∞ln x = ∞.Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 3 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another Definitionex: Inverse of ln xDefinitionThe exp function E (x) = exis theinverse of the log function L(x) = ln x:L ◦E (x) = ln ex= x, ∀x.Propertiesln x is the inverse of ex: ∀x > 0, E ◦ L = eln x= x.∀x > 0, y = ln x ⇔ ey= x.graph(ex) is the reflection of graph(ln x) by line y = x.range(E ) = domain(L) = (0, ∞),domain(E ) = range(L) = (−∞, ∞).limx→−∞ex= 0 ⇔ limx→0+ln x = −∞,limx→∞ex= ∞ ⇔ limx→∞ln x = ∞.Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 3 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another Definitionex: Inverse of ln xDefinitionThe exp function E (x) = exis theinverse of the log function L(x) = ln x:L ◦E (x) = ln ex= x, ∀x.Propertiesln x is the inverse of ex: ∀x > 0, E ◦ L = eln x= x.∀x > 0, y = ln x ⇔ ey= x.graph(ex) is the reflection of graph(ln x) by line y = x.range(E ) = domain(L) = (0, ∞),domain(E ) = range(L) = (−∞, ∞).limx→−∞ex= 0 ⇔ limx→0+ln x = −∞,limx→∞ex= ∞ ⇔ limx→∞ln x = ∞.Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 3 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another Definitionex: Inverse of ln xDefinitionThe exp function E (x) = exis theinverse of the log function L(x) = ln x:L ◦E (x) = ln ex= x, ∀x.Propertiesln x is the inverse of ex: ∀x > 0, E ◦ L = eln x= x.∀x > 0, y = ln x ⇔ ey= x.graph(ex) is the reflection of graph(ln x) by line y = x.range(E ) = domain(L) = (0, ∞),domain(E ) = range(L) = (−∞, ∞).limx→−∞ex= 0 ⇔ limx→0+ln x = −∞,limx→∞ex= ∞ ⇔ limx→∞ln x = ∞.Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 3 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another Definitionex: Inverse of ln xDefinitionThe exp function E (x) = exis theinverse of the log function L(x) = ln x:L ◦E (x) = ln ex= x, ∀x.Propertiesln x is the inverse of ex: ∀x > 0, E ◦ L = eln x= x.∀x > 0, y = ln x ⇔ ey= x.graph(ex) is the reflection of graph(ln x) by line y = x.range(E ) = domain(L) = (0, ∞),domain(E ) = range(L) = (−∞, ∞).limx→−∞ex= 0 ⇔ limx→0+ln x = −∞,limx→∞ex= ∞ ⇔ limx→∞ln x = ∞.Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 3 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another Definitionex: Inverse of ln xDefinitionThe exp function E (x) = exis theinverse of the log function L(x) = ln x:L ◦E (x) = ln ex= x, ∀x.Propertiesln x is the inverse of ex: ∀x > 0, E ◦ L = eln x= x.∀x > 0, y = ln x ⇔ ey= x.graph(ex) is the reflection of graph(ln x) by line y = x.range(E ) = domain(L) = (0, ∞),domain(E ) = range(L) = (−∞, ∞).limx→−∞ex= 0 ⇔ limx→0+ln x = −∞,limx→∞ex= ∞ ⇔ limx→∞ln x = ∞.Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 3 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another DefinitionAlgebraic PropertyLemmaex+y= ex· ey.e−x=1ex.ex−y=exey.erx= (ex)r, ∀r rational.Proofln ex+y= x + y = ln ex+ ln ey= ln (ex· ey) .Since ln x is one-to-one, thenex+y= ex· ey.Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 4 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another DefinitionAlgebraic PropertyLemmaex+y= ex· ey.e−x=1ex.ex−y=exey.erx= (ex)r, ∀r rational.Proofln ex+y= x + y = ln ex+ ln ey= ln (ex· ey) .Since ln x is one-to-one, thenex+y= ex· ey.Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 4 / 17Definition and Properties Differentiation Integration Arbitrary PowersDefinition Properties Another DefinitionAlgebraic PropertyLemmaex+y= ex· ey.e−x=1ex.ex−y=exey.erx= (ex)r, ∀r rational.Proofln ex+y= x + y = ln ex+ ln ey= ln (ex· ey) .Since ln x is one-to-one, thenex+y= ex· ey.Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 4 January 24, 2008 4 / 17Definition and Properties Differentiation Integration Arbitrary