Calculus Examples

$f (x) = x^{3} - x$ , $[0, 4]$

Step 1

If $f$ 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 $(b, f (b))$ .

If $f (x)$ 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 $f (x) = x^{3} - x$ is continuous.

Tap for more steps...

Step 2.1

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:

${x | x \in ℝ}$

Step 2.2

$f (x)$ is continuous on $[0, 4]$ .

The function is continuous.

Step 3

Find the derivative.

Tap for more steps...

Step 3.1

Find the first derivative.

Tap for more steps...

Step 3.1.1

Differentiate.

Tap for more steps...

Step 3.1.1.1

By the Sum Rule, the derivative of $x^{3} - x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x]$

Step 3.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- x]$

Step 3.1.2

Evaluate $\frac{d}{d x} [- x]$ .

Tap for more steps...

Step 3.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$3 x^{2} - \frac{d}{d x} [x]$

Step 3.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 3.1.2.3

Multiply $- 1$ by $1$ .

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

Step 3.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 1$ .

$3 x^{2} - 1$

Step 4

Find if the derivative is continuous on $(0, 4)$ .

Tap for more steps...

Step 4.1

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:

${x | x \in ℝ}$

Step 4.2

$f' (x)$ is continuous on $(0, 4)$ .

The function is continuous.

Step 5

The function is differentiable on $(0, 4)$ because the derivative is continuous on $(0, 4)$ .

The function is differentiable.

Step 6

$f (x)$ satisfies the two conditions for the mean value theorem. It is continuous on $[0, 4]$ and differentiable on $(0, 4)$ .

$f (x)$ is continuous on $[0, 4]$ and differentiable on $(0, 4)$ .

Step 7

Evaluate $f (a)$ from the interval $[0, 4]$ .

Tap for more steps...

Step 7.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{3} - (0)$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - (0)$

Step 7.2.2

Subtract $0$ from $0$ .

$f (0) = 0$

Step 7.2.3

The final answer is $0$ .

$0$

Step 8

Evaluate $f (b)$ from the interval $[0, 4]$ .

Tap for more steps...

Step 8.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = {(4)}^{3} - (4)$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Simplify each term.

Tap for more steps...

Step 8.2.1.1

Raise $4$ to the power of $3$ .

$f (4) = 64 - (4)$

Step 8.2.1.2

Multiply $- 1$ by $4$ .

$f (4) = 64 - 4$

Step 8.2.2

Subtract $4$ from $64$ .

$f (4) = 60$

Step 8.2.3

The final answer is $60$ .

$60$

Step 9

Solve $3 x^{2} - 1 = \frac{- (f (b) + f (a))}{- (b + a)}$ for $x$ . $3 x^{2} - 1 = \frac{- (f (4) + f (0))}{- (4 + 0)}$ .

Tap for more steps...

Step 9.1

Simplify $\frac{(60) - (0)}{(4) - (0)}$ .

Tap for more steps...

Step 9.1.1

Simplify the numerator.

Tap for more steps...

Step 9.1.1.1

Multiply $- 1$ by $0$ .

$3 x^{2} - 1 = \frac{60 + 0}{4 - (0)}$

Step 9.1.1.2

Add $60$ and $0$ .

$3 x^{2} - 1 = \frac{60}{4 - (0)}$

Step 9.1.2

Simplify the denominator.

Tap for more steps...

Step 9.1.2.1

Multiply $- 1$ by $0$ .

$3 x^{2} - 1 = \frac{60}{4 + 0}$

Step 9.1.2.2

Add $4$ and $0$ .

$3 x^{2} - 1 = \frac{60}{4}$

Step 9.1.3

Divide $60$ by $4$ .

$3 x^{2} - 1 = 15$

Step 9.2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 9.2.1

Add $1$ to both sides of the equation.

$3 x^{2} = 15 + 1$

Step 9.2.2

Add $15$ and $1$ .

$3 x^{2} = 16$

Step 9.3

Divide each term in $3 x^{2} = 16$ by $3$ and simplify.

Tap for more steps...

Step 9.3.1

Divide each term in $3 x^{2} = 16$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{16}{3}$

Step 9.3.2

Simplify the left side.

Tap for more steps...

Step 9.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{16}{3}$

Step 9.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{16}{3}$

Step 9.4

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

$x = \pm \sqrt{\frac{16}{3}}$

Step 9.5

Simplify $\pm \sqrt{\frac{16}{3}}$ .

Tap for more steps...

Step 9.5.1

Rewrite $\sqrt{\frac{16}{3}}$ as $\frac{\sqrt{16}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{16}}{\sqrt{3}}$

Step 9.5.2

Simplify the numerator.

Tap for more steps...

Step 9.5.2.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \frac{\sqrt{4^{2}}}{\sqrt{3}}$

Step 9.5.2.2

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

$x = \pm \frac{4}{\sqrt{3}}$

Step 9.5.3

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 9.5.4

Combine and simplify the denominator.

Tap for more steps...

Step 9.5.4.1

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 9.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 9.5.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 9.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{4 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 9.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 9.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 9.5.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 9.5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.5.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{4 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{4 \sqrt{3}}{3^{1}}$

Step 9.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{4 \sqrt{3}}{3}$

Step 9.6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 9.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{4 \sqrt{3}}{3}$

Step 9.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{4 \sqrt{3}}{3}$

Step 9.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{4 \sqrt{3}}{3}, - \frac{4 \sqrt{3}}{3}$

Step 10

There is a tangent line found at $x = \frac{4 \sqrt{3}}{3}$ parallel to the line that passes through the end points $a = 0$ and $b = 4$ .

There is a tangent line at $x = \frac{4 \sqrt{3}}{3}$ parallel to the line that passes through the end points $a = 0$ and $b = 4$

Step 11

There is a tangent line found at $x = - \frac{4 \sqrt{3}}{3}$ parallel to the line that passes through the end points $a = 0$ and $b = 4$ .

There is a tangent line at $x = - \frac{4 \sqrt{3}}{3}$ parallel to the line that passes through the end points $a = 0$ and $b = 4$

Step 12