Stewart 9th Section 3.6: Graphing with Calculus and Technology

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f'\) and \(f''\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x) = x^5 - 5x^4 - x^3 + 28x^2 - 2x\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f'\) and \(f''\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x) = -2x^6 + 5x^5 + 140x^3 - 110x^2\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f'\) and \(f''\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x) = x^6 - 5x^5 + 25x^3 - 6x^2 - 48x\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f'\) and \(f''\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x) = \dfrac{x^4 - x^3 - 8}{x^2 - x - 6}\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f'\) and \(f''\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x) = \dfrac{x}{x^3 + x^2 + 1}\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f'\) and \(f''\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x) = 6 \sin x - x^2\), \(-5 \leq x \leq 3\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f'\) and \(f''\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x) = 6 \sin x + \cot x\), \(-\pi \leq x \leq \pi\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f'\) and \(f''\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x) = \dfrac{\sin x}{x}\), \(-2\pi \leq x \leq 2\pi\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. \(f(x) = 1 + \dfrac{1}{x} + 8/x^2 + 1/x^3\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. \(f(x) = 1/x^8 - (2 \times 10^8)/x^4\)

Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs using a calculator or computer that display the major features of the curve. Use these graphs to estimate the maximum and minimum values. \(f(x) = \dfrac{(x+4)(x-3)^2}{x^4 (x-1)}\)

Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs using a calculator or computer that display the major features of the curve. Use these graphs to estimate the maximum and minimum values. \(f(x) = \dfrac{(2x+3)^2 (x-2)^5}{x^3 (x-5)^2}\)

For the function \(f\) of Example 3, use a computer algebra system to calculate \(f'\) and then graph it to confirm that all the maximum and minimum values are as given in the example. Calculate \(f''\) and use it to estimate the intervals of concavity and inflection points.

For the function \(f\) of Exercise 12, use a computer algebra system to find \(f'\) and \(f''\) and use their graphs to estimate the intervals of increase and decrease and concavity of \(f\).

Use a computer algebra system to graph \(f\) and to find \(f'\) and \(f''\). Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\). \(f(x) = \dfrac{x^3 + 5x^2 + 1}{x^4 + x^3 - x^2 + 2}\)

Use a computer algebra system to graph \(f\) and to find \(f'\) and \(f''\). Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\). \(f(x) = \dfrac{x^{\dfrac{2}{3}}}{1 + x + x^4}\)

Use a computer algebra system to graph \(f\) and to find \(f'\) and \(f''\). Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\). \(f(x) = \sqrt{x + 5 \sin x}\), \(x \leq 20\)

Use a computer algebra system to graph \(f\) and to find \(f'\) and \(f''\). Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\). \(f(x) = \dfrac{2x - 1}{\sqrt[4]{x^4 + x + 1}}\)

In Example 4 we considered a member of the family of functions \(f(x) = \sin(x + \sin c x)\) that occur in FM synthesis. Here we investigate the function with \(c = 3\). Start by graphing \(f\) in the viewing rectangle \([0, \pi]\) by \([-1.2, 1.2]\). How many local maximum points do you see? The graph has more than are visible to the naked eye. To discover the hidden maximum and minimum points you will need to examine the graph of \(f'\) very carefully. In fact, it helps to look at the graph of \(f''\) at the same time. Find all the maximum and minimum values and inflection points. Then graph \(f\) in the viewing rectangle \([-2\pi, 2\pi]\) by \([-1.2, 1.2]\) and comment on symmetry.

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x) = x^3 + c x\)

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x) = x^2 + 6 x + \dfrac{c}{x}\) (trident of Newton)

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x) = x \sqrt{c^2 - x^2}\)

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x) = \dfrac{c x}{1 + c^2 x^2}\)

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x) = \dfrac{\sin x}{c + \cos x}\)

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when \(c\) changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x) = c x + \sin x\)

The figure shows graphs (in blue) of several members of the family of polynomials \(f(x) = c x^4 - 4 x^2 + 1\).

Investigate the family of curves given by the equation \(f(x) = x^4 + c x^2 + x\). Start by determining the transitional value of \(c\) at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There is another transitional value of \(c\) at which the number of critical numbers changes. Try to discover it graphically. Then prove what you have discovered.

Graph the polynomial \(f(x) = 2x^6 + 3x^5 + 3x^3 - 2x^2\). Use the graphs of \(f'\) and \(f''\) to estimate all maximum and minimum points and intervals of concavity.

Draw the graph of the function \(f(x) = \dfrac{x^2 + 7x + 3}{x^2}\) in a viewing rectangle that shows all the important features of the function. Estimate the local maximum and minimum values and the intervals of concavity. Then use calculus to find these quantities exactly.

Graph the function \(f(x) = \dfrac{x^2 (x+1)^3}{(x-2)^2 (x-4)^4}\).

Graph the function \(f(x) = \sin(x + \sin 2x)\). For \(0 \leq x \leq \pi\), estimate all maximum and minimum values, intervals of increase and decrease, and inflection points.

How does the graph of \(f(x) = \dfrac{1}{x^2 + 2x + c}\) vary as \(c\) varies?