Curve Sketching

Scientific Notebook produces excellent function plots, but the default plot may well obscure some of the subtle detail. For example, let us examine the graph of the function f (x) = ${\frac{{x^{6}-5x^{3}+10x^{2}-40x}}{{\left( x^{2}-4\right) ^{2}}}}$. In the default plot, the two vertical asymptotes x = - 2 and x = 2 are visible, but little else is.

$\blacktriangleright$ Plot 2D + Rectangular

${\dfrac{{x^{6}-5x^{3}+10x^{2}-40x}}{{\left( x^{2}-4\right) ^{2}}}}$

dtbpF3in2.0003in0pt

To see more detail, zoom in and experiment with different views such as the following.dtbpF3in2.0003in0pt

To locate the relative extreme values, solve f′(x) = 0.

$\blacktriangleright$ Solve + Exact

f′(x) = 0, Solution is : $\left\{\vphantom{ x=\rho }\right.$x = ρ$\left.\vphantom{ x=\rho }\right\}$

where ρ is a root of 2Z7 -24Z5 +5Z4 +180Z2 -20Z3 - 80Z + 160

You can find approximate real roots of this seventh-degree polynomial with Numeric from the Solve submenu.

$\blacktriangleright$ Solve + Numeric

2Z7 -24Z5 +5Z4 +180Z2 -20Z3 - 80Z + 160 = 0,

Solution is : $\left\{\vphantom{ Z=-3.886400582}\right.$Z = - 3.886400582$\left.\vphantom{ Z=-3.886400582}\right\}$,$\left\{\vphantom{ Z=2.235875042}\right.$Z = 2.235875042$\left.\vphantom{ Z=2.235875042}\right\}$,$\left\{\vphantom{ Z=3.032705128}\right.$Z = 3.032705128$\left.\vphantom{ Z=3.032705128}\right\}$

These three roots of f′ give two local minimums;

f (- 3.886400582) = 32.81201997

and

f (3.032705128) = 22.55314538

and one local maximum,

f (2.235875042) = 29.65559266

You can gain additional insight into this graph by rewriting f (x) as a polynomial plus a fraction.

$\blacktriangleright$ Polynomials + Divide

${\dfrac{{x^{6}-5x^{3}+10x^{2}-40x}}{{\left( x^{2}-4\right) ^{2}}}}$ = x2 +8 + ${\dfrac{{-5x^{3}+58x^{2}-40x-128}}{{\left( x^{2}-4\right) ^{2}}}}$

Select and drag the expression x2 + 8 to the view to see both curves in the same picture. Note how well the graph of y = x2 + 8 matches the graph of y = f (x) for large values of x.

$\blacktriangleright$ Plot 2D + Rectangular

${\dfrac{{x^{6}-5x^{3}+10x^{2}-40x}}{{\left( x^{2}-4\right) ^{2}}}}$, x2 + 8

dtbpF3in2.0003in0pt

To locate the intervals where the graph of f (x) = x4 +3x3 - x2 - 3x is concave upward, evaluate f′′(x) to get f′′(x) = 12x2 + 18x - 2, and solve the inequality 12x2 +18x - 2 > 0.

$\blacktriangleright$ Solve + Exact

12x2 +18x - 2 > 0,

Solution is : $\left\{\vphantom{ x<-\frac{3}{4}-\frac{1}{12}\sqrt{105}}\right.$x < - ${\frac{{3}}{{4}}}$ - ${\frac{{1}}{{12}}}$$\sqrt{{105}}$$\left.\vphantom{ x<-\frac{3}{4}-\frac{1}{12}\sqrt{105}}\right\}$,$\left\{\vphantom{ -\frac{3}{4}+\frac{1}{12}\sqrt{105}<x}\right.$ - ${\frac{{3}}{{4}}}$ + ${\frac{{1}}{{12}}}$$\sqrt{{105}}$ < x$\left.\vphantom{ -\frac{3}{4}+\frac{1}{12}\sqrt{105}<x}\right\}$

To solve more complicated inequalities or systems of inequalities, you can set expressions equal to zero and test for sign changes.


\begin{example} You can answer the question of where the graph of $f(x)=\frac{%... ...2),$\ $(-2,0.1375939264)$, and $(2.341382231,\infty )$.\medskip \end{example}