Notes_Exercises_Set05 - 1. The exponential and logarithmic...

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Unformatted text preview: 1. The exponential and logarithmic functions (Inverse Functions): ) ln( u p e u p = ⇔ = Example: ) 1 ln( 1 = ⇔ = e ) ln( ) ln( u p e u and p e = = Example: 1 ) ln( = e Rules of Exponents Rules of Logarithms 2 1 2 1 2 1 2 1 2 1 2 1 ) ( / p p p p p p p p p p p p e e e e e e e e = = =- + ) ln( 1 ln[ ) ln( ) ln( ] / ln[ ) ln( ) ln( ] ln[ u v u v u v u v u uv v =- = + = 1 = e ) 1 ln( = ) ln( a u u e a = Conversion to base e ) ln( ) ln( log a u u a = Conversion to base e 2. Derivatives x x x e e D = and ) ln( ) ln( ) ln( ) ln( a a a e e D a D x a x a x x x x = = = Anti-derivative: c e dx e x x + = & and c a a dx a x x + = & ) ln( ) exp( * * * ] 1 )[ exp( )] exp( ) [exp( lim ) exp( ] 1 ) [exp( lim ) exp( ] 1 ) )[exp( exp( lim )] exp( ) exp( ) exp( lim ) exp( ) exp( lim ] [ x x x x x x x x x x x x x x x x x x x e D x x x x x x x = = Δ- Δ + = Δ- Δ = Δ- Δ = = Δ- Δ = Δ- Δ + = → Δ → Δ → Δ → Δ → Δ *** since 1 )] exp( ) [exp( lim = Δ- Δ + → Δ x x x because we define exp(x) with a special base = e , such that 1 is the value of the slope of the tangent line at (0,1) for the exponential function....
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Notes_Exercises_Set05 - 1. The exponential and logarithmic...

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