Hyperbolic functions
- 1. 1 2 3 4 65 Hyperbolic functions and Hanging cables.
- 2. Differential and Integral Calculus Course code: MAT101 Section: 03 Group name: BIZSPARK 1. Abdullah Al Hadi 2018-1-80-060 2. Saidur Rahama 2018-1-60-221 3. Premangshu Mondal 2016-2-30-033 4. Shak E Nobat 2018-1-80-026
- 3. HYPERBOLIC FUNCTIONSHYPERBOLIC FUNCTIONS • Vincenzo RiccatiVincenzo Riccati • (1707 - 1775) is(1707 - 1775) is given credit forgiven credit for introducing theintroducing the hyperbolic functions.hyperbolic functions. Hyperbolic functions are very useful in both mathematics and physics.Hyperbolic functions are very useful in both mathematics and physics. In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
- 4. THE HYPERBOLIC FUNCTIONS ARE:THE HYPERBOLIC FUNCTIONS ARE: Hyperbolic sine:Hyperbolic sine: Hyperbolic cosine:
- 5. EQUILATERAL HYPERBOLAEQUILATERAL HYPERBOLA • x = cosh , y = sinhα αx = cosh , y = sinhα α • xx22 – y– y22 = cosh= cosh22 αα- sinh- sinh22 αα= 1.= 1.
- 6. GRAPHS OF HYPERBOLIC FUNCTIONSGRAPHS OF HYPERBOLIC FUNCTIONS • y =y = sinhsinh xx • y =y = coshcosh xx
- 7. THE ST. LOUIS ARCH IS IN THETHE ST. LOUIS ARCH IS IN THE SHAPE OF A HYPERBOLIC COSINE.SHAPE OF A HYPERBOLIC COSINE.
- 9. Y = COSH XY = COSH X •The curve formed by a hanging necklace is called aThe curve formed by a hanging necklace is called a catenary. Its shape follows the curve of . Its shape follows the curve of y = cosh x.y = cosh x.
- 10. CATENARY CURVESCATENARY CURVES • The curve described by a uniform, flexible chain hanging under the influence ofThe curve described by a uniform, flexible chain hanging under the influence of gravity is called agravity is called a catenary curve.catenary curve. This is the familiar curve of an electric wireThis is the familiar curve of an electric wire hanging between two telephone poles.hanging between two telephone poles.
- 11. CATENARY CURVECATENARY CURVE • The curve is described by a COSH(theta) functionThe curve is described by a COSH(theta) function
- 13. GRAPHS OF TANH AND COTH FUNCTIONSGRAPHS OF TANH AND COTH FUNCTIONS • y =y = tanhtanh xx • y =y = coth xcoth x
- 14. GRAPHS OF SINH, COSH, AND TANHGRAPHS OF SINH, COSH, AND TANH
- 15. GRAPHS OF SECH AND CSCH FUNCTIONSGRAPHS OF SECH AND CSCH FUNCTIONS • y =y = sechsech xx • y =y = cschcsch xx
- 16. • Useful relationsUseful relations coshcosh22 x – sinhx – sinh22 x = 1x = 1 Hence:Hence: 1 - (tanh x)1 - (tanh x)22 = (sech x)= (sech x)22 ..
- 17. RELATIONSHIPS OF HYPERBOLIC FUNCTIONSRELATIONSHIPS OF HYPERBOLIC FUNCTIONS • tanh x = sinh x/cosh xtanh x = sinh x/cosh x • coth x = 1/tanh x = cosh x/sinh xcoth x = 1/tanh x = cosh x/sinh x • sech x = 1/cosh xsech x = 1/cosh x • csch x = 1/sinh xcsch x = 1/sinh x • cosh2x - sinh2x = 1cosh2x - sinh2x = 1 • sech2x + tanh2x = 1sech2x + tanh2x = 1 • coth2x - csch2x = 1coth2x - csch2x = 1
- 18. HYPERBOLIC FORMULAS FORHYPERBOLIC FORMULAS FOR INTEGRATIONINTEGRATION -1 2 2 2 2 cosh ln ( - ) - du u C or u u a au a = + + ÷ ∫ 1 2 2 2 2 sinh ln ( ) du u C or u u a aa u − = + + + ÷ + ∫ 1 2 2 1 1 tanh , ln , 2 du u a u C u a or C u a a u a a a a u − + = + < + ≠ ÷ − − ∫
- 19. HYPERBOLIC FORMULAS FOR INTEGRATIONHYPERBOLIC FORMULAS FOR INTEGRATION RELATIONSHIPS OF HYPERBOLIC FUNCTIONS 2 2 1 2 2 1 1 csc ln ( ) , 0. a a udu u h C or C u a a a uu a u − + + =− + − + ≠ + ∫ 2 2 1 2 2 1 1 sec ln ( ) ,0 a a udu u h C or C u a a a a uu a u − + − =− + − + < < − ∫
- 20. ANIMATED PLOT OF THEANIMATED PLOT OF THE TRIGONOMETRICTRIGONOMETRIC (CIRCULAR) AND(CIRCULAR) AND HYPERBOLICHYPERBOLIC FUNCTIONSFUNCTIONS • InIn redred, curve of equation, curve of equation x² + y² = 1 (unit circle),x² + y² = 1 (unit circle), and inand in blueblue,, x² - y² = 1 (equilateral hyperbola),x² - y² = 1 (equilateral hyperbola), with the pointswith the points (cos( ),sin( ))θ θ(cos( ),sin( ))θ θ and (1,tan( )) inθand (1,tan( )) inθ redred andand (cosh( ),sinh( ))θ θ(cosh( ),sinh( ))θ θ and (1,tanh( )) inθand (1,tanh( )) inθ blueblue..
- 21. APPLICATIONS OF HYPERBOLIC FUNCTIONSAPPLICATIONS OF HYPERBOLIC FUNCTIONS • Hyperbolic functions occur in the solutions of some important linearHyperbolic functions occur in the solutions of some important linear differential equations, for example the equation defining a catenary,differential equations, for example the equation defining a catenary, and Laplace's equation in Cartesian coordinates. The latter isand Laplace's equation in Cartesian coordinates. The latter is important in many areas of physics, including electromagnetic theory,important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.heat transfer, fluid dynamics, and special relativity.
- 22. DERIVATIVES OF HYPERBOLIC FUNCTIONSDERIVATIVES OF HYPERBOLIC FUNCTIONS • d/dx(sinh(x)) = cosh(x)d/dx(sinh(x)) = cosh(x) • d/dx(cosh(x)) = - sinh(x)d/dx(cosh(x)) = - sinh(x) • d/dx(tanh(x)) = sechd/dx(tanh(x)) = sech22 (x)(x)
- 23. INTEGRALS OF HYPERBOLIC FUNCTIONSINTEGRALS OF HYPERBOLIC FUNCTIONS • ∫∫ sinh(x)dx = cosh(x) + csinh(x)dx = cosh(x) + c • ∫∫ cosh(x)dx = sinh(x) + c.cosh(x)dx = sinh(x) + c. • ∫∫ tanh(x)dx = ln(cosh x) + c.tanh(x)dx = ln(cosh x) + c.
- 24. EXAMPLE :EXAMPLE : • Find d/dx (sinhFind d/dx (sinh22 (3x))(3x)) Sol: Using the chain rule,Sol: Using the chain rule, we have:we have: d/dx (sinhd/dx (sinh22 (3x))(3x)) = 2 sinh(3x) d/dx (sinh(3x))= 2 sinh(3x) d/dx (sinh(3x)) = 6 sinh(3x) cosh(3x)= 6 sinh(3x) cosh(3x)
- 25. INVERSE HYPERBOLIC FUNCTIONSINVERSE HYPERBOLIC FUNCTIONS • (sinh(sinh 1−1− (x)) =(x)) = • (cosh(cosh 1−1− (x)) =(x)) = .. (tanh(tanh 1−1− (x)) =(x)) = 2 1 1 x− d dx d dx 2 1 1 x+ 2 1 1x − d dx
- 26. CURVES ON ROLLER COASTER BRIDGECURVES ON ROLLER COASTER BRIDGE
- 27. Thank You