Hyperbolics. Scott Morgan. Further Mathematics Support Programme - WJEC A-Level Further Mathematics 31st March scott3142.
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1 Hyperbolics Scott Morgan Further Mathematics Support Programme - WJEC A-Level Further Mathematics 3st March 208
2 Topics Hyperbolic Identities Calculus with Hyperbolics - Differentiation & Integration Inverse Hyperbolic Functions 2
3 Topics Hyperbolic Identities 3
4 Hyperbolic Functions sinh(x) cosh(x) 4
5 Hyperbolic Functions sinh(x) = ex e x 2 cosh(x) = ex + e x 2 4
6 Hyperbolic Functions y = sinh(x) f(x) x 5 0 5
7 Hyperbolic Functions y = cosh(x) f(x) x 6
8 Hyperbolic Functions y = tanh(x) f(x) x 5 7
9 Hyperbolic Functions tanh(x) = sinh(x) cosh(x) cosech(x) = sinh(x) sech(x) = cosh(x) coth(x) = cosh(x) sinh(x) 8
10 Hyperbolic Functions tanh(x) = sinh(x) cosh(x) = ex e x e x + e x cosech(x) = sinh(x) = 2 e x e x sech(x) = cosh(x) = 2 e x + e x coth(x) = cosh(x) sinh(x) = ex + e x e x e x 8
11 Hyperbolic Functions Examples: Prove the following identities, using the exponential definitions of the hyperbolic functions. cosh 2 (x) sinh 2 (x) = cosh(2x) = + 2 sinh 2 (x) cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y) 9
12 Trigonometric vs Hyperbolic Identities Trigonometric Identity Hyperbolic Identity cos 2 (x) + sin 2 (x) = cosh 2 (x) sinh 2 (x) = + tan 2 (x) = sec 2 (x) tanh 2 (x) = sech 2 (x) cot 2 (x) + = cosec 2 (x) coth 2 (x) = cosech 2 (x) 0
13 Trigonometric vs Hyperbolic Identities Trigonometric Identity sin(2x) = 2 sin(x) cos(x) Hyperbolic Identity sinh(2x) = 2 sinh(x) cosh(x) cos(2x) = cos 2 (x) sin 2 (x) cos(2x) = cos 2 (x) + sin 2 (x) cos(2x) = 2 cos 2 (x) cosh(2x) = 2 cosh 2 (x) cos(2x) = 2 sin 2 (x) cosh(2x) = + 2 sinh 2 (x) tan(2x) = 2 tan(x) tan 2 (x) tanh(2x) = 2 tanh(x) +tanh 2 (x)
14 Trigonometric vs Hyperbolic Identities Trigonometric Identity sin(x + y) = sin(x) cos(y) + cos(x) sin(y) Hyperbolic Identity sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y) sin(x y) = sin(x) cos(y) cos(x) sin(y) sinh(x y) = sinh(x) cosh(y) cosh(x) sinh(y) cos(x + y) = cos(x) cos(y) sin(x) sin(y) cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y) cos(x y) = cos(x) cos(y) + sin(x) sin(y) cosh(x y) = cosh(x) cosh(y) sinh(x) sinh(y) 2
15 Equations with Hyperbolic Functions Examples: Solve the following equations: 3 sinh(x) cosh(x) = tanh(x) + 4 sech(x) = 4 2 cosh 2 (x) + 7 sinh(x) = 24 3
16 Topics Calculus with Hyperbolics - Differentiation & Integration 4
17 Calculus with Hyperbolic Functions - Differentiation d(sinh(x)) dx d(cosh(x)) dx 5
18 Calculus with Hyperbolic Functions - Differentiation d(sinh(x)) dx d(cosh(x)) dx = d dx ( e x e x ) = ex + e x 2 2 = cosh(x) 5
19 Calculus with Hyperbolic Functions - Differentiation d(sinh(x)) dx d(cosh(x)) dx ( e x e x ) = ex + e x = d dx 2 2 = d ( e x + e x ) = ex e x dx 2 2 = cosh(x) = sinh(x) 5
20 Calculus with Hyperbolic Functions - Differentiation Examples: Find the derivatives of the following: tanh(x) cosech(x) sech(x) coth(x) 6
21 Calculus with Hyperbolic Functions - Differentiation Examples: Find the derivatives of the following: tanh(2x) sech 2 (x) sinh(4x) cosh 3 (x) x sinh(x) e x sinh(x) cosh(5x) e cosh(x) ln(sinh(x)) 7
22 Calculus with Hyperbolic Functions - Integration sinh(x)dx cosh(x)dx 8
23 Calculus with Hyperbolic Functions - Integration sinh(x)dx = cosh(x) + c cosh(x)dx 8
24 Calculus with Hyperbolic Functions - Integration sinh(x)dx = cosh(x) + c cosh(x)dx = sinh(x) + c 8
25 Calculus with Hyperbolic Functions - Integration Examples: Find the following: x sinh(2x)dx sinh 2 (x)dx cosh 2 (x)dx sinh(3x)dx cosh(5x)dx 3x cosh(4x)dx sinh(x) cosh(x)dx 9
26 Topics Inverse Hyperbolic Functions 20
27 Inverse Hyperbolic Functions Example: Using the exponential definition for sinh(x), show that ( sinh (x) = ln x + ) x 2 + ( cosh (x) = ln x + ) x 2 tanh (x) = 2 ln ( x + x ) 2
28 Inverse Hyperbolic Functions y = sinh (x) f(x) x 5 22
29 Inverse Hyperbolic Functions y = cosh (x) f(x) x 23
30 Inverse Hyperbolic Functions y = tanh (x) f(x) x 5 24
31 Inverse Hyperbolic Functions Example: Using implicit differentiation, find: d ( sinh (x) ) dx d ( cosh (x) ) dx 25
32 Inverse Hyperbolic Functions Example: Using implicit differentiation, find: d ( sinh (x) ) = dx + x 2 d ( cosh (x) ) dx 25
33 Inverse Hyperbolic Functions Example: Using implicit differentiation, find: d ( sinh (x) ) = dx d ( cosh (x) ) = dx + x 2 x 2 25
34 Standard Integrals ( a2 x = x ) 2 sin ( x < a) a a 2 + x 2 = ( x ) a tan a ( x 2 a = x ) 2 cosh = ln (x + ) x a 2 a 2 ( x < a) ( x 2 + a = x ) 2 sinh = ln (x + ) x a 2 + a 2 a 2 x 2 = ( x ) a tanh = a 2a ln a + x a x ( x < a) x 2 a 2 = 2a ln x a x + a 26
35 Calculus with Hyperbolic Functions - Integration Examples: Find the following: dx 6 + x x 2 dx x 2 25 dx 9 x 2 dx dx 9 x 2 x 2 25 dx 27
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