Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.3.12

Chapter 7, Problem 7.3.12

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.

tanh(−x) = −tanh x

Verified step by step guidance

Step 1: Recall the definition of the hyperbolic tangent function:

\tanh (x) = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}

Step 2: Substitute

- x

into the definition of

\tanh

. This gives:

\tanh (- x) = \frac{e^{- x} - e^{x}}{e^{- x} + e^{x}}

Step 3: Factor out

- 1

from the numerator of the fraction:

\tanh (- x) = \frac{- (e^{x} - e^{- x})}{e^{- x} + e^{x}}

Step 4: Recognize that the numerator is the negative of the numerator in the definition of

\tanh (x)

. Thus,

\tanh (- x) = - \tanh (x)

Step 5: Conclude that the identity

\tanh (- x) = - \tanh (x)

is proven using the definition of hyperbolic functions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are based on hyperbolas rather than circles. The primary hyperbolic functions include sinh(x), cosh(x), and tanh(x), which are defined as sinh(x) = (e^x - e^(-x))/2, cosh(x) = (e^x + e^(-x))/2, and tanh(x) = sinh(x)/cosh(x). Understanding these definitions is crucial for proving identities involving hyperbolic functions.

Odd and Even Functions

An odd function is defined by the property f(-x) = -f(x), while an even function satisfies f(-x) = f(x). The hyperbolic tangent function, tanh(x), is an odd function, which means that tanh(-x) = -tanh(x). This property is essential for proving identities involving tanh, as it allows for simplifications when substituting negative arguments.

Proof Techniques in Calculus

Proof techniques in calculus often involve direct substitution, algebraic manipulation, and the application of definitions. To prove identities like tanh(−x) = −tanh x, one typically starts with the left-hand side, applies the definition of tanh, and uses properties of exponents and odd functions to arrive at the right-hand side. Mastery of these techniques is vital for successfully demonstrating mathematical identities.

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11–15. Identities Prove each identity using the definitions of the hyperbolic functions.tanh(−x) = −tanh x

Verified video answer for a similar problem:

Key Concepts

Hyperbolic Functions

Odd and Even Functions

Proof Techniques in Calculus

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.

tanh(−x) = −tanh x