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Questions and Answers
Given $x \in (-1, 1)$, which of the following expressions is equivalent to $\tanh^{-1}(x)$?
Given $x \in (-1, 1)$, which of the following expressions is equivalent to $\tanh^{-1}(x)$?
- $\frac{1}{2} \log(\frac{1+x}{1-x})$ (correct)
- $\frac{1}{2} \log(\frac{1-x}{1+x})$
- $\log(x - \sqrt{x^2 + 1})$
- $\log(x + \sqrt{x^2 + 1})$
Which hyperbolic function is defined as $\frac{1}{2}(e^x - e^{-x})$?
Which hyperbolic function is defined as $\frac{1}{2}(e^x - e^{-x})$?
- $\sinh(x)$ (correct)
- $\text{sech}(x)$
- $\tanh(x)$
- $\cosh(x)$
Which of the following is the correct definition of $\tanh(x)$?
Which of the following is the correct definition of $\tanh(x)$?
- $\frac{\cosh(x)}{\sinh(x)}$
- $\frac{1}{\sinh(x) \cdot \cosh(x)}$
- $\frac{\sinh(x)}{\cosh(x)}$ (correct)
- $\sinh(x) \cdot \cosh(x)$
Given that $\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$, which expression is equivalent to $\tanh(x)$?
Given that $\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$, which expression is equivalent to $\tanh(x)$?
Which function represents $\cosh(x)$?
Which function represents $\cosh(x)$?
If $\tanh(x) = y$, express $x$ in terms of $y$.
If $\tanh(x) = y$, express $x$ in terms of $y$.
Given the definitions of $\sinh(x)$ and $\cosh(x)$, which of the following identities is correct?
Given the definitions of $\sinh(x)$ and $\cosh(x)$, which of the following identities is correct?
How is $\tanh(x)$ related to $\sinh(x)$ and $\cosh(x)$?
How is $\tanh(x)$ related to $\sinh(x)$ and $\cosh(x)$?
If $f(x) = \tanh(x)$, what is the range of $f(x)$?
If $f(x) = \tanh(x)$, what is the range of $f(x)$?
Simplify the expression: $\frac{\cosh(2x) + 1}{\sinh(2x)}$
Simplify the expression: $\frac{\cosh(2x) + 1}{\sinh(2x)}$
Flashcards
tanh(x)
tanh(x)
It is defined as sinh(x) / cosh(x), analogous to tan(x) = sin(x) / cos(x).
sinh(x)
sinh(x)
This is one half times (e to the x, minus e to the minus x).
cosh(x)
cosh(x)
It is expressed as one half times (e to the x, plus e to the minus x).
tanh⁻¹(x)
tanh⁻¹(x)
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tanh⁻¹(x) formula
tanh⁻¹(x) formula
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Study Notes
- Correct answer is option 3: 1/2 log((1+x)/(1-x))
- 30% of people got it right
Concepts
- tanh(x) = sinh(x) / cosh(x)
- sinh(x) = 1/2 * (e^x - e^-x)
- cosh(x) = 1/2 * (e^x + e^-x)
Solution
- tanh(x) is also equal to (e^x - e^-x) / (e^x + e^-x) or (e^2x - 1) / (e^2x + 1)
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