Calculus

James Stewart

Chapter 3 Differentiation Rules - all with Video Answers

Educators

Section 11

Hyperbolic Functions

00:24

Problem 1

Find the numerical value of each expression.
(a) $ \sin h 0 $
(b) $ \cos h 0 $

Amrita B.

Numerade Educator

00:33

Problem 2

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

Amrita B.

Numerade Educator

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Problem 3

Find the numerical value of each expression.
(a) $ \cosh (\ln 5) $
(b) $ \cosh 5 $

Leon D.

Numerade Educator

00:37

Problem 4

Find the numerical value of each expression.
(a) $ \sinh 4 $
(b) $ \sinh (\ln 4) $

Amrita B.

Numerade Educator

00:32

Problem 5

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

Amrita B.

Numerade Educator

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Problem 6

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

Ma. Theresa A.

Numerade Educator

02:18

Problem 7

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

Yuki H.

Numerade Educator

01:57

Problem 8

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

Yuki H.

Numerade Educator

00:36

Problem 9

Prove the identity.
$ \cosh x + \sinh x = e^x $

Amrita B.

Numerade Educator

00:28

Problem 10

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

Amrita B.

Numerade Educator

00:52

Problem 11

Prove the identity.
$ \sinh (x + y ) = \sinh x \cosh y + \cosh x \sinh

Amrita B.

Numerade Educator

00:50

Problem 12

Prove the identity.
$ \cosh (x + y) = \cosh x \cosh y + \sinh x \sinh y $

Amrita B.

Numerade Educator

03:39

Problem 13

Prove the identity.
$ \coth^2 x - 1 = csch^2 x $

Carson M.

Numerade Educator

01:02

Problem 14

Prove the identity.
$ \tanh (x + y) = \frac { \tanh x + \tanh y}{1 + \tanh x \tanh y} $

Amrita B.

Numerade Educator

00:39

Problem 15

Prove the identity.
$ \sinh 2x = 2 \sinh x \cosh x $

Amrita B.

Numerade Educator

00:42

Problem 16

Prove the identity.
$ \cosh 2x = \cosh^2 x + \sinh^2 x $

Amrita B.

Numerade Educator

01:09

Problem 17

Prove the identity.
$ \tanh (\ln x) = \frac {x^2 - 1}{x^2 + 1} $

Amrita B.

Numerade Educator

01:10

Problem 18

Prove the identity.
$ \frac {1 + \tanh x}{1 - \tanh x} = e^{2x} $

Amrita B.

Numerade Educator

00:59

Problem 19

Prove the identity.
$ (\cosh x + \sinh x)^n = \cosh nx + \sinh nx $ ($ n $ any real number)

Amrita B.

Numerade Educator

00:49

Problem 20

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

Amrita B.

Numerade Educator

00:48

Problem 21

If $ \cosh x = \frac {5}{3} $ and $ x > 0, $ find the values of the other hyperbolic functions at $ x. $

Amrita B.

Numerade Educator

03:24

(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 device to produce them.

Clarissa N.

Numerade Educator

10:33

Problem 23

Use the definitions of the hyperbolic functions find each of the following limits.
(a) $ \displaystyle \lim_{x \to ^x} \tanh x $
(b) $ \displaystyle \lim_{x \to ^{-x}} \tanh x $
(c) $ \displaystyle \lim_{x \to ^x} \sinh x $
(d) $ \displaystyle \lim_{x \to ^{-x}} \sinh x $
(e) $ \displaystyle \lim_{x \to ^x} sech x $
(f) $ \displaystyle \lim_{x \to ^x} \coth x $
(g) $ \displaystyle \lim_{x \to 0^+} \coth x $ (h) $ \displaystyle \lim_{x \to 0^-} \coth x $
(i) $ \displaystyle \lim_{x \to ^{-x}} csch x $
(j) $ \displaystyle \lim_{x \to ^x} \frac {\sinh x}{e^x} $

Clarissa N.

Numerade Educator

10:04

Problem 24

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

Clarissa N.

Numerade Educator

00:54

Problem 25

Give an alternative solution to Example 3 by letting $ y = \sinh^{-1} x $ and then using Exercise 9 and Example 1(a) with $ x $ replaced by $ y. $

Amrita B.

Numerade Educator

00:43

Problem 26

Prove Equation 4.

Amrita B.

Numerade Educator

00:56

Problem 27

Prove Equation 5 using (a) the method of Example 3 and (b) Exercise 18 with $ x $ replaced by $ y. $

Amrita B.

Numerade Educator

02:41

Problem 28

For each of the following function (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3.
(a) $ csch^{-1} $
(b) $ sech^{-1} $
(c) $ \coth^{-1} $

Linda H.

Numerade Educator

06:42

Problem 29

Prove the formulas given in Table 6 for the derivatives of the following functions.
(a) $ \cosh^{-1} $
(b) $ \tanh^{-1} $
(c) $ csch^{-1} $
(d) $ sech^{-1} $
(e) $ \coth^{-1} $

Clarissa N.

Numerade Educator

00:51

Problem 30

Find the derivative. Simplify where possible.
$ f(x) = e^x \cosh x $

Amrita B.

Numerade Educator

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Problem 31

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

Suman Saurav T.

Numerade Educator

00:43

Problem 32

Find the derivative. Simplify where possible.
$ g(x) = \sinh^2 x $

Amrita B.

Numerade Educator

00:25

Problem 33

Find the derivative. Simplify where possible.
$ h(x) = \sinh (x^2) $

Amrita B.

Numerade Educator

00:33

Problem 34

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

Amrita B.

Numerade Educator

00:32

Problem 35

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

Amrita B.

Numerade Educator

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Problem 36

Find the derivative. Simplify where possible.
$ y = sech x (1 + \ln sech x) $

Suman Saurav T.

Numerade Educator

00:32

Problem 37

Find the derivative. Simplify where possible.
$ y = e^{\cosh 3x} $

Amrita B.

Numerade Educator

00:46

Problem 38

Find the derivative. Simplify where possible.
$ f(t) = \frac {1 + \sinh t}{1 - \sinh t} $

Amrita B.

Numerade Educator

01:00

Problem 39

Find the derivative. Simplify where possible.
$ g(t) = t \coth \sqrt {t^2 + 1} $

Amrita B.

Numerade Educator

00:26

Problem 40

Find the derivative. Simplify where possible.
$ y = \sinh^{-1} (\tan x) $

Amrita B.

Numerade Educator

03:49

Problem 41

Find the derivative. Simplify where possible.
$ y = \cosh^{-1} \sqrt x $

Yuki H.

Numerade Educator

00:29

Problem 42

Find the derivative. Simplify where possible.
$ y = x \tanh^{-1}x + \ln \sqrt {1 - x^2} $

Amrita B.

Numerade Educator

00:42

Problem 43

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

Amrita B.

Numerade Educator

00:30

Problem 44

Find the derivative. Simplify where possible.
$ y = sech^{-1} (e^{-x}) $

Amrita B.

Numerade Educator

00:34

Problem 45

Find the derivative. Simplify where possible.
$ y = \coth^{-1} (\sec x) $

Amrita B.

Numerade Educator

00:47

Problem 46

Show that $ \frac {d}{dx} \sqrt[4]{ \frac {1 + \tanh x}{1 - \tanh x}} = \frac {1}{2} e^{x/2}. $

Amrita B.

Numerade Educator

00:47

Problem 47

Show that $ \frac {d}{dx} \arctan (\tan x) = sech 2x. $

Amrita B.

Numerade Educator

15:36

Problem 48

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 $ \mid x \mid \le 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)?

Jessica W.

University of Missouri - Columbia

00:49

Problem 49

If a water wave with length $ L $ moves with velocity $ v $ in a body of water with depth $ d, $ then
$ v = \sqrt {\frac {gL}{2 \pi} \tanh (\frac {2 \pi d}{L}} $
where $ g $ is the acceleration due to gravity. (See Figure 5.) Explain why the approximation
$ v \approx \sqrt {\frac {gL}{2 \pi}} $ is appropriate in deep water.

Amrita B.

Numerade Educator

02:53

Problem 50

A flexible cable always hangs in the shape of a centenary $ y = c + a \cosh (x/a), $ where $ c $ and $ a $ are constants and $ a > 0 $ (see Figure 4 and Exercise 52.) Graph several members of the family of functions $ y = a \cosh (x/a). $ How does the graph change as $ a $ varies?

Yuki H.

Numerade Educator

04:57

Problem 51

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 pole.
(b) Find the angle $ \theta $ between the line and the pole.

Avantika G.

Numerade Educator

01:03

Problem 52

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
$ \frac {d^2 y}{dx^2} = \frac {pg}{T} \sqrt {1 + (\frac {dy}{dx})^2} $
where $ p $ 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 coordinates system is chosen appropriately. Verify that the function
$ y = f(x) = \frac {T}{pg} \cosh (\frac {pgx}{T}) $
is a solution of this differential equation.

Amrita B.

Numerade Educator

01:34

Problem 53

A cable with linear density $ p = kg/m $ is strung from the tops of two poles that are $ 200 m $ apart.
(a) Use Exercise 52 to find the tension $ T $ so that 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 poles now?

Amrita B.

Numerade Educator

04:20

Problem 54

A model for the velocity of a falling object after time $ t $ is
$ v(t) = \sqrt {\frac {mg}{k}} \tanh (t \sqrt { \frac {gk}{m}}) $
where $ m $ is the mass of the object, $ g = 9.8 m/s^2 $ is the acceleration due to gravity, $ k $ is a constant, $ t $ is measured in seconds, and $ v $ in $ m/s. $
(a) Calculate the terminal velocity of the objects, that is, $ \lim_{t \to ^x} v(t). $
(b) If a person falls from a building, 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 falls in belly-to-earth position, what is the terminal velocity? What about feet-first?

Mutahar M.

Numerade Educator

05:36

Problem 55

(a) Show that any function of the form
$ y = A \sinh mx + B \cosh mx $ satisfies the differential equation $ y" = m^2 y. $
(b) Find $ y = y(x) $ such that $ y" = 9y, y(0) = -4, $ and $ y'(0) = 6. $

Clarissa N.

Numerade Educator

00:31

Problem 56

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

Amrita B.

Numerade Educator

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Problem 57

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

Saad Ali K.

Numerade Educator

03:04

Problem 58

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

Yuki H.

Numerade Educator

00:31

Problem 59

Show that if $ a \not= 0 $ and $ b \not= 0, $ then there exist numbers $ \alpha $ and $ \beta $ such that $ ae^x + be^{-x} $ equals either
$ \alpha \sinh (x + \beta) $ or $ \alpha \cosh (x + \beta) $
In other words, almost every function of the form $ f(x) = ae^e + be^{-x} $ is a shifted and stretched hyperbolic sine or cosine function.

Amrita B.

Numerade Educator