$$\dfrac{1}{2} \tanh(2x - 7) + C$$

$$\dfrac{1}{2} \tanh(2x - 7) + C$$

Question 1

Which of the following is the correct derivative of $$y = \$$sinh^2$$ 6x$$ with respect to $$x$$?

Accepted Answer

$$12 \sinh 6x \cosh 6x$$

Question 2

A medical researcher models the growth of a tumor using $$y = \ln(\sinh 7x). $$What is the rate of change of the tumor size with respect to $$x$$?

Accepted Answer

$$7 \coth 7x$$

Question 3

Find the derivative of $$y = \$$cosh^{-1}$$ 2x + 9$$ with respect to $$x.$$

Accepted Answer

$$\dfrac{2}{\sqrt{4x^2$ - 1}}$$

Question 4

A chemical reaction is modeled by $$y = (1 - 3t) \$$coth^{-1}$$ 3t. $$What is $$\dfrac{dy}{dt}$$?

Accepted Answer

$$-3 \$$coth^{-1}$$ 3t$$

Question 5

Evaluate $$\int \sinh(4x - \ln 4) \, dx.$$

Accepted Answer

$$\dfrac{1}{2} \cosh(4x - \ln 4) + C$$

Question 6

A population model uses the function $$f(x) = \$$sech^2$$(2x - 7). $$What is $$\int \$$sech^2$$(2x - 7) \, dx$$?

Accepted Answer

$$\dfrac{1}{2} 	anh(2x - 7) + C$$

Question 7

Evaluate $$\int_{\ln 2}^{\ln 7} 	anh x \, dx as it $$might appear in a financial model for exponential growth.

Accepted Answer

$$\ln 20$$

Question 8

A physicist is analyzing the integral $$\$$int_0$$^{\ln 2} e$$^{-t}$$ \sinh t \, dt. $$What is the value of this integral?

Accepted Answer

$$\dfrac{\ln 2}{2} - \dfrac{3}{16}$$

Question 9

Evaluate $$\$$int_5$$^6 \dfrac{dx}{25 + x^2$} in $$terms of an inverse hyperbolic function, as might be used in pharmacokinetics.

Accepted Answer

$$\$$sinh^{-1}$$\left(\dfrac{6}{5}\right) - \$$sinh^{-1}$$\left(\dfrac{5}{5}\right)$$

Question 10

Evaluate $$\$$int_4$$^8 \dfrac{dx}{x^2$ - 9} in $$terms of natural logarithms, as might be used in a radioactive decay calculation.

Accepted Answer

$$\ln 8 + \dfrac{55}{4} + \dfrac{7}{11}$$

Question 11

Evaluate $$\$$int_0$$^{3\Delta} \dfrac{-\sin x}{1 + \cos^2$ x} \, dx as it $$might appear in a signal processing context.

Accepted Answer

$$\ln$$^{-1}$$ + \dfrac{2}{3}$$

Question 12

Which of the following best describes the relationship between $$\sinh x$$ and $$\cosh x$$?

Accepted Answer

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

Question 13

A biologist models cell growth with $$y = \ln(\sinh 7x). If$$ $$x$$ increases by 0.1 units, what is the approximate change in $$y$$ near $$x = 1$$?

Accepted Answer

$$0.7 \coth 7x$$

Question 14

Which integral would be most appropriate to model the total effect of a drug over time if the drug's concentration follows $$e^{-t}$$ \sinh t$$?

Accepted Answer

$$\$$int_0$$^{\ln 2} e$$^{-t}$$ \sinh t \, dt$$

Question 15

A signal is modeled by $$f(x) = \$$sech^2$$(2x - 7). $$Which of the following is the correct antiderivative?

Accepted Answer

$$\dfrac{1}{2} 	anh(2x - 7) + C$$

Question 16

Which of the following is the correct formula for the derivative of $$y = \$$cosh^{-1}$$ x$$?

Accepted Answer

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

Question 17

A financial analyst uses $$\int_{\ln 2}^{\ln 7} 	anh x \, dx to $$estimate growth. What is the result?

Accepted Answer

$$\ln 20$$

Question 18

Which of the following is the correct expression for $$\int \dfrac{dx}{a^2$ + x^2$}$$?

Accepted Answer

$$\dfrac{1}{a} \$$sinh^{-1}$$\left(\dfrac{x}{a}\right) + C$$

Step-by-Step Guidance for Calculus II: Hyperbolic Functions, Derivatives, and Integrals

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