Questions: Hyperbolic Functions

Question 1

What is the derivative of cosh x?

• A. -sinh x
• B. sinh x
• C. cosh x
• D. -cosh x

Answer: (B)

cosh x = (eˣ + e⁻ˣ)/2, so its derivative is (eˣ - e⁻ˣ)/2 = sinh x. Unlike cosine, there is no minus sign, because the derivative of e⁻ˣ is -e⁻ˣ.

Question 2

For every real number x, cosh²x - sinh²x = 1.

• T. True
• F. False

Answer: True

cosh²x = (e²ˣ + 2 + e⁻²ˣ)/4 and sinh²x = (e²ˣ - 2 + e⁻²ˣ)/4. Subtracting gives 4/4 = 1.

Question 3

Evaluate cosh 0 and sinh 0, and explain why cosh x ≥ 1 for all real x.

Model answer: cosh 0 = 1 and sinh 0 = 0. Since cosh²x = 1 + sinh²x ≥ 1 and cosh x is positive, cosh x ≥ 1.

At x = 0, e⁰ = 1, so cosh 0 = (1 + 1)/2 = 1 and sinh 0 = (1 - 1)/2 = 0. The identity forces cosh²x ≥ 1, and cosh x is a sum of positive terms divided by 2, so cosh x ≥ 1.

Question 4

Which curve describes the shape of a flexible cable hanging under its own weight between two supports?

• A. A parabola y = ax²
• B. A catenary y = a cosh(x/a)
• C. A sine curve y = a sin(x)
• D. A circle

Answer: (B)

A uniform hanging cable forms a catenary, which is a scaled hyperbolic cosine. It resembles a parabola near its lowest point but is a different curve.