 # Hyperbolic Functions.

## Presentation on theme: "Hyperbolic Functions."— Presentation transcript:

Hyperbolic Functions

The Hyperbolic Sine, Hyperbolic Cosine & Hyperbolic Tangent

The Inverse Hyperbolic Cotangent, Hyperbolic Secant & Hyperbolic Cosecant

Values

Remember Logarithmic Identities ( Calculus I )

Logarithmic Identities II

Identities

Proofs Identity (1)

Proofs Identity (2)

Proofs Identity (3)

Proofs Identity (4)

Proofs Identity (5)

Proofs Identity (6)

Proofs Identity (7)

Derivatives of Hyperbolic Functions

Proofs (1) (sinhx)’ = coshx (2) (coshx)’ = sinhx

Proofs (3) (tanhx)’ = sech2x

Proofs (4) (cothx)’ = -csch2x

Proofs (5) (sechx)’ = - sechx tanhx

Proofs (5) (cschx)’ = - cschx cothx

Integrals Involving hyperbolic Functions

Examples I

Examples II

Examples III

Examples IV

Examples V

Graphs of Hyperbolic Functions

Graphs of Exponential Functions Functions

Graphs of Exponential Functions

f(x) = Cosh x

f(x) = cosh x = (½)ex + (½) e-x as x → ∞ the values f(x) → ∞ following (½)ex as x → - ∞ the values f(x) → ∞ following (½) e-x

f(x) = sinh x

f(x) = tanh x

f(x) = secx