Q: Which of the following is the correct definition of the hyperbolic tangent function?

$$\tanh x = \dfrac{\sinh x}{\cosh x}$$

Question 1

Which of the following is the correct definition of the hyperbolic sine function?

Accepted Answer

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

Question 2

The equation $$y = a \cosh\left(\dfrac{x}{a}\right)$$ describes the shape of which real-world structure?

Accepted Answer

A catenary (hanging cable or rope)

Question 3

Evaluate $$\cosh(\ln 2)$$ without a calculator.

Accepted Answer

Question 4

Which identity is true for all $$x$$?

Accepted Answer

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

Question 5

Find the derivative of $$f(x) = \cosh(8x + 1).$$

Accepted Answer

$$8 \sinh(8x + 1)$$

Question 6

Which of the following integrals is equal to $$	anh u + C$$?

Accepted Answer

$$\int \$$sech^2$$ u\ du$$

Question 7

Evaluate $$\int \$$sech^2$$(5x)\ dx.$$

Accepted Answer

$$\dfrac{1}{5} 	anh(5x) + C$$

Question 8

Which of the following statements about the graphs of hyperbolic functions is correct?

Accepted Answer

$$\cosh x is an $$even function and $$\sinh x is an $$odd function.

Question 9

Which of the following is the correct formula for the arc length $$L of a $$curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$L = \int_a^b \sqrt{1 + [f'(x)]^2}\ dx$$

Question 10

Which of the following integrals equals $$\ln|\sinh x| + C$$?

Accepted Answer

$$\int \coth x\ dx$$

Question 11

A climber anchors a rope at two points of equal height, separated by 100 ft, and the rope follows $$f(x) = 200 \cosh\left(\dfrac{x}{200}\right)$$ for $$x \in [-50, 50]. $$Which integral gives the length of the rope?

Accepted Answer

$$\int_{-50}^{50} \sqrt{1 + [f'(x)]^2}\ dx$$

Question 12

Which of the following is the correct derivative of $$\sech u$$ with respect to $$x$$, where $$u is a $$differentiable function of $$x$$?

Accepted Answer

$$-(\sech u 	anh u) u'$$

Question 13

Which of the following is the correct definition of the hyperbolic tangent function?

Accepted Answer

$$	anh x = \dfrac{\sinh x}{\cosh x}$$

Question 14

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

Accepted Answer

$$\ln \cosh x + C$$

Question 15

If $$\cosh x = 1$$, what is the value of $$x$$?

Accepted Answer

$$x = 0$$

Question 16

Which of the following is the correct formula for $$\sinh(2x)$$?

Accepted Answer

$$2 \sinh x \cosh x$$

Question 17

Which of the following is the correct formula for $$\$$cosh^2$$ x in $$terms of $$\cosh 2x$$?

Accepted Answer

$$\$$cosh^2$$ x = \dfrac{1 + \cosh 2x}{2}$$

Question 18

Which of the following is the correct formula for $$\int \sech x\ dx$$?

Accepted Answer

$$	an$$^{-1}$$(\sinh x) + C$$

Question 19

Which of the following is the correct formula for $$\int \csch x\ dx$$?

Accepted Answer

$$\ln |	anh(x/2)| + C$$

Hyperbolic Functions: Definitions, Properties, Calculus, and Applications

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