Q: Which of the following is the correct definition of the hyperbolic tangent function $$\tanh x$$?

$$\tanh x = \dfrac{e^x - e$$^{-x}$$}{e^x + e$$^{-x}$$}$$

Question 1

Which of the following is the correct definition of the hyperbolic sine function $$\sinh x$$?

Accepted Answer

$$\sinh x = \dfrac{e^x - e$$^{-x}$$}{2}$$

Question 2

The Gateway Arch in St. Louis was designed using which hyperbolic function?

Accepted Answer

$$\cosh x$$

Question 3

Which identity is true for all $$x$$ regarding hyperbolic functions?

Accepted Answer

$$\$$cosh^2$$ x - \$$sinh^2$$ x = 1$$

Question 4

What is the derivative of $$y = \sinh x$$ with respect to $$x$$?

Accepted Answer

$$\dfrac{d}{dx} \sinh x = \cosh x$$

Question 5

Which of the following is the logarithmic form of the inverse hyperbolic sine function $$\$$sinh^{-1}$$ x$$?

Accepted Answer

$$\$$sinh^{-1}$$ x = \ln(x + \sqrt{x^2$ + 1})$$

Question 6

If $$y = \cosh x$$, what is $$\dfrac{d}{dx} y$$?

Accepted Answer

$$\sinh x$$

Question 7

Which of the following integrals leads to an inverse hyperbolic function?

Accepted Answer

$$\int \dfrac{1}{\sqrt{x^2$ + 1}} dx$$

Question 8

A radioactive substance decays according to $$y = y_0$ e$$^{-kt}$$. If the half-life is T$, what is the value of k$?

Accepted Answer

$$k = \dfrac{\ln 2}{T}$$

Question 9

Which differential equation models Newton's Law of Cooling?

Accepted Answer

$$\dfrac{dH}{dt} = k(H - H_s)$$

Question 10

Which of the following is the correct definition of the hyperbolic tangent function $$	anh x$$?

Accepted Answer

$$	anh x = \dfrac{e^x - e$$^{-x}$$}{e^x + e$$^{-x}$$}$$

Question 11

Suppose the atmospheric pressure $$p at $$altitude $$h$$ satisfies $$\dfrac{dp}{dh} = kp. If$$ $$p_0 is $$the pressure at sea level ($$h=0$$), what is the general solution for $$p$$?

Accepted Answer

$$p = p_0$ e$$^{kh}$$

Question 12

Which of the following is a separable differential equation?

Accepted Answer

$$\dfrac{dy}{dx} = y \cos x$$

Question 13

Evaluate $$\int \$$sech^2$$ x \, dx.$$

Accepted Answer

$$	anh x + C$$

Question 14

If $$y = \$$tanh^{-1}$$ x$$, what is $$\dfrac{dy}{dx}$$ for $$|x| \u003c 1$$?

Accepted Answer

$$\dfrac{1}{1 - x^2$}$$

Question 15

Which of the following is the correct integral formula for $$\int \sinh x \, dx$$?

Accepted Answer

$$\cosh x + C$$

Question 16

A population grows according to $$y = y_0$ e$$^{kt}$$. If the population doubles in 5 years, what is the value of k$?

Accepted Answer

$$k = \dfrac{\ln 2}{5}$$

Question 17

Which of the following is the correct definition of the hyperbolic secant function $$\sech x$$?

Accepted Answer

$$\sech x = \dfrac{1}{\cosh x}$$

Question 18

Solve the separable differential equation $$\dfrac{dy}{dx} = y^2$ with initial condition y(0) = 1$. What is y(x$$)$$?

Accepted Answer

$$y(x) = \dfrac{1}{1 - x}$$

Question 19

Which of the following is the correct identity for $$\cosh 2x$$?

Accepted Answer

$$\cosh 2x = \$$cosh^2$$ x + \$$sinh^2$$ x$$

Hyperbolic Functions, Exponential Change, and Separable Differential Equations

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