\( \tanh (1) \)

The function \( \tanh(x) \) is the hyperbolic tangent function. To calculate \( \tanh(1) \), we can use the following approximation: \[ \tanh(x) \approx \frac{e^x - e^{-x}}{e^x + e^{-x}} \] Substituting \( x = 1 \) into the approximation, we get: \[ \tanh(1) \approx \frac{e^1 - e^{-1}}{e^1 + e^{-1}} \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(\frac{e^{1}-e^{-1}}{e^{1}+e^{-1}}\) - step1: Evaluate the power: \(\frac{e-e^{-1}}{e^{1}+e^{-1}}\) - step2: Evaluate the power: \(\frac{e-e^{-1}}{e+e^{-1}}\) - step3: Evaluate the power: \(\frac{\frac{e^{2}-1}{e}}{e+e^{-1}}\) - step4: Evaluate the power: \(\frac{\frac{e^{2}-1}{e}}{\frac{e^{2}+1}{e}}\) - step5: Multiply by the reciprocal: \(\frac{e^{2}-1}{e}\times \frac{e}{e^{2}+1}\) - step6: Reduce the numbers: \(\left(e^{2}-1\right)\times \frac{1}{e^{2}+1}\) - step7: Multiply: \(\frac{e^{2}-1}{e^{2}+1}\) The value of \( \tanh(1) \) is approximately 0.761594.