5. Graphical Applications of Derivatives

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

Plot a possible graph of f.

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) = (x⁴/2) - 3x² + 4x + 1

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) = 3x/(x² + 3)

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = 3x/(x² - 1)

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x²/(x - 2)

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.

x³ + y³ = 3xy (Folium of Descartes)

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.

x⁴ - x² + y² = 0 (Figure-8 curve)

{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.

a. Plot a graph of the curve when a = 3.

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.

{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.

a. Determine the marginal cost and average cost functions. Graph and interpret these functions.

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = (x² + 12)/(2x + 1)

Use the guidelines of this section to make a complete graph of f.

f(x) = tan⁻¹ (x²/√3)

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x⁴ + 4x³

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x³ - 6x² - 135x

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) = ln( x² + 3) / (x -1)