6.7.1 Defining the Hyperbolic Functions

Question 1

Defining the Hyperbolic Functions

Answer 1

• Hanging cables are called catenary curves, and are described by hyperbolic cosine.
• There are six hyperbolic functions with names similar to those of the trig functions. They are defined by adding or subtracting exponential functions.

Question 2

note

Answer 2

• Cables, chains, ropes, and electrical wires – when hanging between two poles – are examples of catenary curves. These curves are described by the hyperbolic cosine function, cosh x.
• Hyperbolic cosine is actually defined as the sum of two
  exponential functions, e x /2 and e –x /2.
• By studying the graphs of these two functions, you can
  generate the graph of cosh x.
• Right now it may not make sense why this function has a trigonometric name when it only involves exponentials.
  Professor Burger will discuss the reasons later.
• The “h” means that these are hyperbolic functions.
• Remember that cosh x involves a sum, while sinh x involves a difference.
• The other four hyperbolic functions follow from the
  definitions of hyperbolic sine and hyperbolic cosine, just like the trig functions are defined by sine and cosine.
• Next up: Why do these functions have trigonometric-like names? And what’s hyperbolic about them?

Question 3

Which of the following hyperbolic functions is depicted in the graph?

Answer 3

sech x

Question 4

Which of the following statements about hyperbolic functions is not correct?

Answer 4

cschx= 1/sinhx= 1/e^x−e^−x

Question 5

Which of the following lists all of the hyperbolic functions that are not defined at x = 0?

Answer 5

coth x and csch x

Question 6

The semicircle in Figure 1 involves the function y=√1−x^2. The hyperbolic region in Figure 2 involves a related function, y=√1+x^2. Which of the following are expressions for the related circle and hyperbola?

Answer 6

circle: x ^2 + y ^2 = 1
hyperbola: −x ^2 + y ^2 = 1

Question 7

Which of the following shows the correct expressions for cosh x and sinh x ?

Answer 7

coshx=e^x+e^−x/2, sinhx=e^x−e^−x/2

Question 8

Which of the following statements about hyperbolic functions is not correct?

Answer 8

Not all of the hyperbolic functions are defined in terms of exponential functions.

Question 9

Which of these graphs does not match its label?

Answer 9

This is the graph of y = −(e ^−x / 2).

Question 10

Which of the following hyperbolic functions is depicted in the graph?

Answer 10

csch x

Question 11

Which of the following hyperbolic functions is depicted in the graph?

Answer 11

tanh x

Question 12

Which of the following are correct expressions for tanh x and coth x ?

Answer 12

tanhx=e^x−e^−x/e^x+e^−x, cothx=e^x+e^−x/e^x−e^−x,x≠0