Question

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 10x² / (x² + 3)

Accepted Answer

Identify the domain of the function. Since the function is a rational function, it is defined for all real numbers except where the denominator is zero. Here, the denominator $$x^2 + 3 is $$never zero, so the domain is all real numbers. Find the intercepts. For the y-intercept, set $$x = 0$$ and solve $$f(0) = \frac{10(0)^2}{0^2 + 3} = 0. $$For x-intercepts, set $$f(x) = 0$$ and solve $$10x^2 = 0$$, which gives $$x = 0. $$Thus, the only intercept is at the origin (0,0). Determine the asymptotic behavior. Since the degree of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients, $$y = \frac{10}{1} = 10. $$There are no vertical asymptotes as the denominator is never zero. Analyze the first derivative $$f'(x) to $$find critical points and determine intervals of increase and decrease. Use the quotient rule: $$f'(x) = \frac{d}{dx}\left(\frac{10x^2}{x^2 + 3}\right). $$Simplify and solve $$f'(x) = 0 to $$find critical points. Examine the second derivative $$f''(x) to $$determine concavity and points of inflection. Use the derivative of $$f'(x)$$ and solve $$f''(x) = 0 to $$find points of inflection. Analyze the sign of $$f''(x) to $$determine concavity on intervals.

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 10x² / (x² + 3)

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Concetti chiave

Function Behavior

Critical Points and Derivatives

Graphing Techniques

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.ƒ(x) = 10x² / (x² + 3)

Risposta video verificata per un problema simile:

Concetti chiave

Function Behavior

Critical Points and Derivatives

Graphing Techniques

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 10x² / (x² + 3)