AP Calculus AB: 6.7.1 Defining the Hyperbolic Functions
This section introduces the six hyperbolic functions—analogous to trigonometric functions—defined using combinations of exponential functions. It explains how real-world shapes like hanging cables (catenaries) are modeled using hyperbolic cosine and covers the basic definitions, graphs, and key properties of functions like cosh, sinh, tanh, and their reciprocals.
Defining the Hyperbolic Functions
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.
Key Terms
Defining the Hyperbolic Functions
Hanging cables are called catenary curves, and are described by hyperbolic cosine.
There are six hyperbolic functions with n...
note
Cables, chains, ropes, and electrical wires – when hanging between two poles – are examples of catenary curves. These curves are described ...
Which of the following hyperbolic functions is depicted in the graph?
sech x
Which of the following statements about hyperbolic functions is not correct?
cschx= 1/sinhx= 1/e^x−e^−x
Which of the following lists all of the hyperbolic functions that are not defined at x = 0?
coth x and csch x
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?
circle: x ^2 + y ^2 = 1
hyperbola: −x ^2 + y ^2 = 1
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| Term | Definition |
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Defining the Hyperbolic Functions |
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note |
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Which of the following hyperbolic functions is depicted in the graph? | sech x |
Which of the following statements about hyperbolic functions is not correct? | cschx= 1/sinhx= 1/e^x−e^−x |
Which of the following lists all of the hyperbolic functions that are not defined at x = 0? | coth x and csch x |
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? | circle: x ^2 + y ^2 = 1 |
Which of the following shows the correct expressions for cosh x and sinh x ? | coshx=e^x+e^−x/2, sinhx=e^x−e^−x/2 |
Which of the following statements about hyperbolic functions is not correct? | Not all of the hyperbolic functions are defined in terms of exponential functions. |
Which of these graphs does not match its label? | This is the graph of y = −(e ^−x / 2). |
Which of the following hyperbolic functions is depicted in the graph? | csch x |
Which of the following hyperbolic functions is depicted in the graph? | tanh x |
Which of the following are correct expressions for tanh x and coth x ? | tanhx=e^x−e^−x/e^x+e^−x, cothx=e^x+e^−x/e^x−e^−x,x≠0 |