FAC1004 Tutorial 8 — Hyperbolic Functions

Practice problems on hyperbolic functions, their derivatives, and integrals.

Topics Covered

• Evaluating hyperbolic functions
• Hyperbolic identities
• Derivatives of hyperbolic functions
• Integrals of hyperbolic functions
• Logarithmic form of inverse hyperbolic functions

Problem Set

1. Evaluate from sinh: Given $\sinh x = \frac{5}{12}$, find:

   • $\cosh x$
   • $\text{csch } x$
   • $\tanh x$
   • $\sinh(2x)$
   • $\cosh(5x)$

2. Solve for x: Show that $x = \frac{1}{2}\ln 3$ if $\tanh x = \frac{1}{2}$

3. Hyperbolic Equation: Find all possible values of $\sinh x$ for which $12\cosh^2 x + 7\sinh x = 24$

4. Basic Derivatives: Find $\frac{dy}{dx}$ for:

   • $y = \sinh(4x - 8)$
   • $y = \cosh(x^4)$
   • $y = \coth(\ln x)$
   • $y = \ln(\tanh(2x))$
   • $y = \sinh^3(2x)$

5. Advanced Derivatives: Differentiate:

   • $y = x^4\cosh(3x^2 - 8x)$
   • $y = \cosh(\sin^{-1}(x^4))$
   • $y = \coth(\ln x)\cos^{-1}(2x^3)$
   • $y = \frac{\ln(\sin(2x))}{\coth(x^2)}$
   • $y = \frac{\sinh^3(2x)}{\cosh^2(5x^{-3}) + x}$

6. Basic Integrals: Evaluate:

   • $\int \sinh x \cosh(x) , dx$
   • $\int \cosh(2x - 3) , dx$

7. Inverse Hyperbolic Values: Evaluate:

   • $\tanh^{-1} 0$
   • $\cosh^{-1} \pi$
   • $\sinh^{-1}\frac{\pi}{4}$

8. Prove Logarithmic Form: Prove $\sinh^{-1} x = \ln\left[x + \sqrt{1+x^2}\right]$

9. Solve Inverse Hyperbolic Equations: Find $x$ for various inverse hyperbolic equations

Related

• FAC1004 - Advanced Mathematics II (Computing) — main course page
• Hyperbolic Functions — concept page
• FAC1004 L17 — Hyperbolic Functions — related lecture
• FAC1004 L18 — Hyperbolic Functions (Derivatives & Integrals) — related lecture

Source File

TUTORIALS_SET_2526/FAC1004 Tutorial 8 25-26.pdf