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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 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

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Step 1: Recall the definition of the hyperbolic tangent function: tanh(x)=ex-e-xex+e-x.
Step 2: Substitute -x into the definition of tanh. This gives: tanh(-x)=e-x-exe-x+ex.
Step 3: Factor out -1 from the numerator of the fraction: tanh(-x)=-(ex-e-x)e-x+ex.
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.
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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.
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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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Related Practice
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Textbook Question

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∫ cosh 2x dx

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Newton’s method Use Newton’s method to find all local extreme values of ƒ(x) = x sech x.

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Textbook Question

22–36. Derivatives Find the derivatives of the following functions.


f(t) = 2 tanh⁻¹ √t

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Textbook Question

Uranium dating Uranium-238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of U-238 and lead, and they determine that 85% of the original U-238 remains; the other 15% has decayed into lead. How old is the rock?

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