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Cusps, inflection points and tangents

Exercises · Derivatives · with worked solutions · PDF

Exercise 1

Using the definition of derivative, determine the behavior at the origin of the following function:

\[ f(x)=x^{\frac{4}{3}} \]
Solution

Let

\[ f(x) = x^{\frac{4}{3}} \]

then for \(x_0=0\)

\[ \frac{f(h) - f(0)}{h} = \frac{ h^{4/3}}{h} = h^{1/3} \]

and the limits are:

\[ \lim_{h \rr 0^-} {h^{1/3}} = 0 {\rm ~~~and~~~}\lim_{h \rr 0^+} {h^{1/3}} = 0 {\rm ~~~~hence~~~~} \lim_{h \rr 0} {h^{1/3}} = 0 {\rm ~~~and~~~}f'(0)= 0. \]

Figure 1

The function has a point with horizontal tangent at \(x_0=0\). The function:

\[ f(h) = h^{\frac{1}{3}} \]

is a power with positive rational exponent \(\frac{m}{n}\) with \(n\) odd (hence defined on all of \(\R\)) and \(m\) odd (hence an odd function). Its graph is:

Figure 2

Exercise 2

Using the definition of derivative, determine the behavior at the origin of the following function:

\[ f(x)=x^{\frac{2}{3}} \]
Solution

Let

\[ f(x) = x^{\frac{2}{3}} \]

then for \(x_0=0\)

\[ \frac{f(h) - f(0)}{h} = \frac{ h^{2/3}}{h} = \frac{ 1}{h^{1/3}} \]

and the limits are:

\[ \lim_{h \rr 0^-} \frac{ 1}{h^{1/3}} = \im {\rm ~~and~~}\lim_{h \rr 0^+} \frac{ 1}{h^{1/3}} = \ip {\rm ~~~~~hence~~~~~} f'_-(0)= \im {\rm ~~and~~}f'_+(0)= \ip \]

Figure 3

The function has a cusp at \(x_0=0\). The function:

\[ f(h) = h^{-\frac{1}{3}} = \frac{1}{h^{{1}/{3}}} \]

is a power with negative rational exponent \(\frac{m}{n}\) with \(n\) odd (hence defined on all of \(\R\)) and \(m\) odd (hence an odd function). Its graph is:

Figure 4

Exercise 3

Using the definition of derivative, determine the behavior at the origin of the following function:

\[ f(x)=x^{\frac{5}{2}} \]
Solution

Let

\[ f(x) = x^{\frac{5}{2}} \]

then for \(x_0=0\)

\[ \frac{f(h) - f(0)}{h} = \frac{ h^{5/2}}{h} = h^{3/2} \]

and the limits are:

\[ \lim_{h \rr 0^+} {h^{3/2}} = 0. \]

Figure 5

The function has a point with horizontal tangent at \(x_0=0\). The function:

\[ f(h) = h^{\frac{3}{2}} \]

is a power with positive rational exponent \(\frac{m}{n}\) with \(n\) even (hence defined only on \(\R_+\)). Its graph is:

Figure 6

Exercise 4

Using the definition of derivative, determine the behavior at the origin of the following function:

\[ f(x)=x^{\frac{3}{2}} \]
Solution

Let

\[ f(x) = x^{\frac{3}{2}} \]

then for \(x_0=0\)

\[ \frac{f(h) - f(0)}{h} = \frac{ h^{3/2}}{h} = h^{1/2} \]

and the limits are:

\[ \lim_{h \rr 0^+} {h^{1/2}} = 0. \]

Figure 7

The function has a point with horizontal tangent at \(x_0=0\). The function:

\[ f(h) = h^{\frac{1}{2}} \]

is a power with positive rational exponent \(\frac{m}{n}\) with \(n\) even (hence defined only on \(\R_+\)). Its graph is:

Figure 8