5. Graphical Applications of Derivatives

Sketch the graphs of the rational functions in Exercises 53–60.

𝓍²

y = ------------------

𝓍² ― 4

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

f(x) = tan⁻¹ (x²/√3)

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 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)

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

y = 1 / (x² - 1)

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.

{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ

Local max/min of x¹⸍ˣ Use analytical methods to find all local extrema of the function ƒ(x) = x¹⸍ˣ , for x > 0 . Verify your work using a graphing utility.

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) = 4cos (π (x-1)) on [0, 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.

x⁴ - x² + y² = 0 (Figure-8 curve)

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

82. y' = sin t, for 0 ≤ t ≤ 2π

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

36. y = (x³ + x - 2) / (x - x²)

{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.

y = 8/(x² + 4) (Witch of Agnesi)

Mean Value Theorem and graphs Find all points on the interval (1,3) at which the slope of the tangent line equals the average rate of change of f on [1,3]. Reconcile your results with the Mean Value Theorem. <IMAGE>

In Exercises 49–52, graph each function. Then use the function’s first derivative to explain what you see.

y = 𝓍²/³ + (𝓍―1)²/³ 

{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.