Calculus Examples

$f (x) = x^{\frac{1}{3}}$ ,

$[- 1, 1]$

Step 1

$f$ is continuous on the interval

$[a, b]$ and differentiable on

$(a, b)$ , then at least one real number

$c$ exists in the interval

$(a, b)$ such that

$f' (c) = \frac{f (b) - f a}{b - a}$ . The mean value theorem expresses the relationship between the slope of the tangent to the curve at

$x = c$ and the slope of the line through the points

$(a, f (a))$ and

$(b, f (b))$ .

$f (x)$ is continuous on

$[a, b]$

and if

$f (x)$ differentiable on

$(a, b)$ ,

then there exists at least one point,

$c$ in

$[a, b]$ :

$f' (c) = \frac{f (b) - f a}{b - a}$ .

Step 2

Check if

$f (x) = x^{\frac{1}{3}}$ is continuous.

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Step 2.1

To find whether the function is continuous on

$[- 1, 1]$ or not, find the domain of

$f (x) = x^{\frac{1}{3}}$ .

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Step 2.1.1

Convert expressions with fractional exponents to radicals.

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Step 2.1.1.1

Apply the rule

$x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt[3]{x^{1}}$

Step 2.1.1.2

Anything raised to

$1$ is the base itself.

$\sqrt[3]{x}$

Step 2.1.2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2.2

$f (x)$ is continuous on

$[- 1, 1]$ .

The function is continuous.

Step 3

Find the derivative.

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Step 3.1

Find the first derivative.

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Step 3.1.1

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = \frac{1}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} - 1}$

Step 3.1.2

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}$

Step 3.1.3

Combine

$- 1$ and

$\frac{3}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply

$- 1$ by

$3$ .

$\frac{1}{3} x^{\frac{1 - 3}{3}}$

Step 3.1.5.2

Subtract

$3$ from

$1$ .

$\frac{1}{3} x^{\frac{- 2}{3}}$

Step 3.1.6

Move the negative in front of the fraction.

$\frac{1}{3} x^{- \frac{2}{3}}$

Step 3.1.7

Simplify.

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Step 3.1.7.1

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}}$

Step 3.1.7.2

Multiply

$\frac{1}{3}$ by

$\frac{1}{x^{\frac{2}{3}}}$ .

$f' (x) = \frac{1}{3 x^{\frac{2}{3}}}$

Step 3.2

The first derivative of

$f (x)$ with respect to

$x$ is

$\frac{1}{3 x^{\frac{2}{3}}}$ .

$\frac{1}{3 x^{\frac{2}{3}}}$

Step 4

Find if the derivative is continuous on

$(- 1, 1)$ .

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Step 4.1

To find whether the function is continuous on

$(- 1, 1)$ or not, find the domain of

$f' (x) = \frac{1}{3 x^{\frac{2}{3}}}$ .

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Step 4.1.1

Apply the rule

$x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{3 \sqrt[3]{x^{2}}}$

Step 4.1.2

Set the denominator in

$\frac{1}{3 \sqrt[3]{x^{2}}}$ equal to

$0$ to find where the expression is undefined.

$3 \sqrt[3]{x^{2}} = 0$

Step 4.1.3

Solve for

$x$ .

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Step 4.1.3.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${(3 \sqrt[3]{x^{2}})}^{3} = 0^{3}$

Step 4.1.3.2

Simplify each side of the equation.

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Step 4.1.3.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{x^{2}}$ as

$x^{\frac{2}{3}}$ .

${(3 x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 4.1.3.2.2

Simplify the left side.

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Step 4.1.3.2.2.1

Simplify

${(3 x^{\frac{2}{3}})}^{3}$ .

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Step 4.1.3.2.2.1.1

Apply the product rule to

$3 x^{\frac{2}{3}}$ .

$3^{3} {(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 4.1.3.2.2.1.2

Raise

$3$ to the power of

$3$ .

$27 {(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 4.1.3.2.2.1.3

Multiply the exponents in

${(x^{\frac{2}{3}})}^{3}$ .

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Step 4.1.3.2.2.1.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 4.1.3.2.2.1.3.2

Cancel the common factor of

$3$ .

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Step 4.1.3.2.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 4.1.3.2.2.1.3.2.2

Rewrite the expression.

$27 x^{2} = 0^{3}$

Step 4.1.3.2.3

Simplify the right side.

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Step 4.1.3.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$27 x^{2} = 0$

Step 4.1.3.3

Solve for

$x$ .

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Step 4.1.3.3.1

Divide each term in

$27 x^{2} = 0$ by

$27$ and simplify.

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Step 4.1.3.3.1.1

Divide each term in

$27 x^{2} = 0$ by

$27$ .

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 4.1.3.3.1.2

Simplify the left side.

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Step 4.1.3.3.1.2.1

Cancel the common factor of

$27$ .

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Step 4.1.3.3.1.2.1.1

Cancel the common factor.

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Step 4.1.3.3.1.2.1.2

Divide

$x^{2}$ by

$1$ .

$x^{2} = \frac{0}{27}$

Step 4.1.3.3.1.3

Simplify the right side.

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Step 4.1.3.3.1.3.1

Divide

$0$ by

$27$ .

$x^{2} = 0$

Step 4.1.3.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 4.1.3.3.3

Simplify

$\pm \sqrt{0}$ .

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Step 4.1.3.3.3.1

Rewrite

$0$ as

$0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 4.1.3.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 4.1.3.3.3.3

Plus or minus

$0$ is

$0$ .

$x = 0$

Step 4.1.4

The domain is all values of

$x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4.2

$f' (x)$ is not continuous on

$(- 1, 1)$ because

$0$ is not in the domain of

$f' (x) = \frac{1}{3 x^{\frac{2}{3}}}$ .

The function is not continuous.

Step 5

The function is not differentiable on

$(- 1, 1)$ because the derivative

$\frac{1}{3 x^{\frac{2}{3}}}$ is not continuous on

$(- 1, 1)$ .

The function is not differentiable.

Step 6