Assume that f is a differentiable function. Find the elasticity of the following: a. x^5 f(x) b. (f(x))^(3/2)

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Understand the Problem

The question is asking to find the elasticity of two mathematical expressions involving a differentiable function f. The user wants to determine how the change in the function is related to the change in its variables for these specific cases.

Answer

a. \( E = \frac{5}{x} + \frac{f'(x)}{f(x)} \) b. \( E = \frac{3}{2} \frac{f'(x)}{f(x)} \cdot x \)
Answer for screen readers

a. The elasticity of ( x^5 f(x) ) is ( E = \frac{5}{x} + \frac{f'(x)}{f(x)} ).

b. The elasticity of ( (f(x))^{3/2} ) is ( E = \frac{3}{2} \frac{f'(x)}{f(x)} \cdot x ).

Steps to Solve

  1. Understanding Elasticity Elasticity measures how a function responds to changes in its variables. For a function ( g(x) ), the elasticity ( E ) is given by: $$ E = \frac{d(g)}{d(x)} \cdot \frac{x}{g(x)} $$

  2. Finding Elasticity of ( x^5 f(x) ) Let ( g(x) = x^5 f(x) ).

    • Differentiate ( g(x) ) using the product rule: $$ g'(x) = 5x^4 f(x) + x^5 f'(x) $$

    • Now compute elasticity: $$ E = \left(5x^4 f(x) + x^5 f'(x)\right) \cdot \frac{x}{x^5 f(x)} = \frac{(5x^4 f(x) + x^5 f'(x)) \cdot x}{x^5 f(x)} $$ Simplifying this gives: $$ E = \frac{5}{x} + \frac{f'(x)}{f(x)} $$

  3. Finding Elasticity of ( (f(x))^{3/2} ) Let ( h(x) = (f(x))^{3/2} ).

    • Differentiate ( h(x) ) using the chain rule: $$ h'(x) = \frac{3}{2}(f(x))^{1/2} \cdot f'(x) $$

    • Now compute elasticity: $$ E = \left(\frac{3}{2}(f(x))^{1/2} f'(x)\right) \cdot \frac{x}{(f(x))^{3/2}} = \frac{3}{2} \frac{f'(x)}{f(x)} \cdot x $$

a. The elasticity of ( x^5 f(x) ) is ( E = \frac{5}{x} + \frac{f'(x)}{f(x)} ).

b. The elasticity of ( (f(x))^{3/2} ) is ( E = \frac{3}{2} \frac{f'(x)}{f(x)} \cdot x ).

More Information

Elasticity helps in understanding how sensitive a function is to changes in its input. In economic terms, it reflects how demand responds to price changes.

Tips

  • Forgetting to apply the product or chain rule correctly when differentiating.
  • Not simplifying the final elasticity expression.

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