?A model for the velocity of a falling object after time t isv(t) = where m is the mass

Find the numerical value of each expression. (a) \(sinh \ 0\) (b) \(cosh \ 0\) Equation Transcription: Text Transcription: sinh 0 cosh 0

Find the numerical value of each expression. (a) tanh 0 (b) tanh 1

Find the numerical value of each expression. (a) \(cosh(ln \ 5)\) (b) \(cosh \ 5\) Equation Transcription: Text Transcription: cosh(ln 5) cosh 5

Find the numerical value of each expression. (a) \(sinh \ 4\) (b) \(sinh(ln \ 4)\) Equation Transcription: Text Transcription: sinh 4 sinh(ln 4)

Find the numerical value of each expression. (a) sech 0 (b) \(cosh^{-1} 1\)

Find the numerical value of each expression. (a) sinh 1 (b) \(\sinh ^{-1} 1\)

Write \(8 \sinh x+5 \cosh x\) in terms of \(e^{x}\) and \(e^{-x}\). Equation Transcription: Text Transcription: 8 sinh x + 5 cosh x e^x e^-x.

Write \(2 e^{2 x}+3 e^{-2 x}\) in terms of sinh 2x and cosh 2x.

Write \(sinh(ln \ x)\) as a rational function of \(x\). Equation Transcription: Text Transcription: x sinh (ln x)

Write \(cosh(4 ln x)\) as a rational function of \(x\) . Equation Transcription: Text Transcription: cosh(4 ln x) x

Prove the identity. sinh (-x) = -sinh x (This shows that sinh is an odd function.)

Prove the identity. cosh(-x) = cosh x (This shows that cosh is an even function.)

\(\cosh \ x+\sinh x=e^{x}\). Equation Transcription: Text Transcription: cosh x + sinh x = e^x

Prove the identity. \(\cosh x-\sinh x=e^{-x}\)

\(\sinh (x+y)=\sinh \ x \ \cosh y+\cosh \ x \sinh \ y\). Equation Transcription: Text Transcription: sinh(x + y) = sinh x cosh y + cosh x sinh y.

cosh(x + y) = cosh x cosh y + sinh x sinh y.

coth2x - 1 = csch2x.

tanh(x+y) =

sinh 2x = 2 sinh x cosh x.

cosh 2x = cosh2x + sinh2x

tanh(ln x) =

(cosh x + sinh x)n = cosh nx + sinh nx (n any real number)

If \(\tanh x=\frac{12}{13}\), find the values of the other hyperbolic functions at x.

If cosh x = and x > 0, find the values of the other hyperbolic functions at x.

(a) Use the graphs of sinh, cosh, and tanh in Figures 1–3 to draw the graphs of csch, sech, and coth. (b) Check the graphs that you sketched in part (a) by using a graphing calculator or computer to produce them.

Use the definitions of the hyperbolic functions to find each of the following limits. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Prove the formulas given in Table 1 for the derivatives of the functions (a) cosh, (b) tanh, (c) csch, (d) sech, and (e) coth.

Give an alternative solution to Example 3 by letting \(y=\sinh ^{-1} x\) and then using Exercise 13 and Example 1 (a) with x replaced by y.

Prove Equation 4.

Prove Equation 5 using (a) the method of Example 3 and (b) Exercise 22 with x replaced by y.

For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3. 1. csch-1 2. sech-1 3. coth-1

Prove the formula given in Table 6 for the derivative of each of the following functions. 1. cosh-1 2. tanh-1 3. coth-1

Prove the formula given in Table 6 for the derivative of each of the following functions. 1. sech-1 2. csch-1

Find the derivative. Simplify where possible. f(x) = cosh 3x

Find the derivative. Simplify where possible. f(x) = ex cosh x

Find the derivative. Simplify where possible. h(x) = sinh(x2)

Find the derivative. Simplify where possible. g(x) = sinh2x

Find the derivative. Simplify where possible. G(t) = sinh(ln t)

Find the derivative. Simplify where possible. F(t) = ln(sinh t)

Find the derivative. Simplify where possible. f(x) = tanh

Find the derivative. Simplify where possible. H(v) = etanh 2v

Find the derivative. Simplify where possible. y = sech x tanh x

Find the derivative. Simplify where possible. y = sech(tanh x)

Find the derivative. Simplify where possible. g(t) = t coth

Find the derivative. Simplify where possible. f(t) =

Find the derivative. Simplify where possible. \(f(x)=\sinh ^{-1}(-2 x)\)

Find the derivative. Simplify where possible. g(x) = tanh-1(x3)

Find the derivative. Simplify where possible. y = cosh-1(sec ????), 0 ? ???? < ????/2

Find the derivative. Simplify where possible. y = sech-1(sin ????), 0 < ???? < ????/2

Find the derivative. Simplify where possible. G(u) = cosh-1 u > 0

Find the derivative. Simplify where possible. y = x tanh-1x + ln

Find the derivative. Simplify where possible. \(y=x \sinh ^{-1}(x / 3)-\sqrt{9+x^{2}}\)

Show that

Show that \(\frac{d}{d x} \arctan (\tanh x)=\operatorname{sech} 2 x\)

The Gateway Arch The Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equation y = 211.49 - 20.96 cosh 0.03291765x for the central curve of the arch, where x and y are measured in meters and | x | ? 91.20. (a) Graph the central curve. (b) What is the height of the arch at its center? (c) At what points is the height 100 m? (d) What is the slope of the arch at the points in part (c)?

If a water wave with length L moves with velocity v in a body of water with depth d, then where g is the acceleration due to gravity. (See Figure 5.) Explain why the approximation is appropriate in deep water.

A flexible cable always hangs in the shape of a catenary y = c + a cosh(x/a), where c and a are constants and a > 0 (see Figure 4 and Exercise 60). Graph several members of the family of functions y = a cosh(x/a). How does the graph change as a varies?

A telephone line hangs between two poles 14 m apart in the shape of the catenary y = 20 cosh(x/20) - 15, where x and y are measured in meters. (a) Find the slope of this curve where it meets the right-hand pole. (b) Find the angle between the line and the pole.

Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve y = f(x) that satisfies the differential equation where is the linear density of the cable, g is the acceleration due to gravity, T is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function y = f(x) = is a solution of this differential equation.

A cable with linear density \(\rho=2 \mathrm{~kg} / \mathrm{m}\) is strung from the tops of two poles that are 200 m apart. (a) Use Exercise 60 to find the tension T so that the cable is 60 m above the ground at its lowest point. How tall are the poles? (b) If the tension is doubled, what is the new low point of the cable? How tall are the poles now?

A model for the velocity of a falling object after time t is v(t) = where m is the mass of the object, g = 9.8 m/s2 is the acceleration due to gravity, k is a constant, t is measured in seconds, and v in m/s. 1. Calculate the terminal velocity of the object, that is, limt v(t). 2. If a person skydives from a plane, the value of the constant k depends on his or her position. For a “belly-to-earth” position, k = 0.515 kg/s, but for a “feet-first” position, k = 0.067 kg/s. If a 60-kg person descends in belly-to-earth position, what is the terminal velocity? What about feet-first? 3. Source: L. Long et al., “How Terminal Is Terminal Velocity?” American Mathematical Monthly 113 (2006): 752–55.

1. Show that any function of the form y = A sinh mx + B cosh mx satisfies the differential equation y” ? m2y. 2. Find y = y(x) such that y” ? 9y, y(0) = -4, and y’(0) = 6.

If \(x=\ln (\sec \theta+\tan \theta)\), show that \(sec \ \theta = cosh \ x\).

At what point of the curve y = cosh x does the tangent have slope 1?

Investigate the family of functions fn(x) = tanh(n sin x) where n is a positive integer. Describe what happens to the graph of fn when n becomes large.

Show that if a 0 and b 0, then there exist numbers and such that aex + be-x equals either sinh(x + ) or cosh(x + ) In other words, almost every function of the form f(x) = aex + be-x is a shifted and stretched hyperbolic sine or hyperbolic cosine function.