Evaluate: [tex]\tanh 4 x[/tex]
Answer (1)
Express tanh 4 x using the double angle formula: tanh 4 x = 1 + t a n h 2 2 x 2 t a n h 2 x , where tanh 2 x = 1 + t a n h 2 x 2 t a n h x .
Alternatively, use the exponential definition: tanh 4 x = e 4 x + e − 4 x e 4 x − e − 4 x .
Simplify the exponential form to: tanh 4 x = e 8 x + 1 e 8 x − 1 .
If x = 1 , then tanh 4 ≈ 0.9993 .
Explanation
Problem Analysis We are asked to evaluate tanh 4 x . We can approach this using the double angle formula for hyperbolic tangents or the exponential definition.
Double Angle Formula Method 1: Using the double angle formula. Recall the double angle formula for hyperbolic tangent: tanh 2 x = 1 + t a n h 2 x 2 t a n h x . We can rewrite tanh 4 x as tanh ( 2 ( 2 x )) . Applying the double angle formula, we get tanh 4 x = tanh ( 2 ( 2 x )) = 1 + tanh 2 2 x 2 tanh 2 x Now, we apply the double angle formula again to express tanh 2 x in terms of tanh x :
tanh 2 x = 1 + tanh 2 x 2 tanh x Substitute this expression into the equation for tanh 4 x :
tanh 4 x = 1 + ( 1 + t a n h 2 x 2 t a n h x ) 2 2 ( 1 + t a n h 2 x 2 t a n h x ) = 1 + ( 1 + t a n h 2 x ) 2 4 t a n h 2 x 1 + t a n h 2 x 4 t a n h x = ( 1 + tanh 2 x ) 2 + 4 tanh 2 x 4 tanh x ( 1 + tanh 2 x ) tanh 4 x = 1 + 2 tanh 2 x + tanh 4 x + 4 tanh 2 x 4 tanh x + 4 tanh 3 x = 1 + 6 tanh 2 x + tanh 4 x 4 tanh x + 4 tanh 3 x This expresses tanh 4 x in terms of tanh x .
Exponential Definition Method 2: Using the exponential definition. Recall the exponential definition of tanh x : tanh x = e x + e − x e x − e − x . Therefore, tanh 4 x = e 4 x + e − 4 x e 4 x − e − 4 x We can rewrite this as: tanh 4 x = e 4 x + e − 4 x e 4 x − e − 4 x = e 4 x + e 4 x 1 e 4 x − e 4 x 1 = e 8 x + 1 e 8 x − 1 Multiplying both numerator and denominator by e − 4 x , we get: tanh 4 x = e 4 x + e − 4 x e 4 x − e − 4 x This is the simplest form using the exponential definition.
Numerical Approximation Without a specific value for x , we cannot evaluate tanh 4 x to a numerical value. However, if we assume x = 1 , then tanh 4 = e 4 + e − 4 e 4 − e − 4 ≈ 54.598 + 0.018 54.598 − 0.018 ≈ 54.616 54.58 ≈ 0.9993 .
Final Answer Thus, tanh 4 x = e 4 x + e − 4 x e 4 x − e − 4 x . If x = 1 , then tanh 4 ≈ 0.9993 .
Examples
In physics, the hyperbolic tangent function appears in various contexts, such as describing the velocity profile of a fluid in laminar flow or modeling the magnetization of a material as a function of an applied magnetic field. Evaluating tanh 4 x for different values of x can help understand how these physical quantities change under varying conditions. For instance, in fluid dynamics, it might represent how quickly the fluid's velocity reaches its maximum value as you move away from a boundary.
