Calculus Examples

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

Find the x-intercepts.

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To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 2 x^{3} + 1$

Solve the equation.

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Rewrite the equation as $2 x^{3} + 1 = 0$ .

$2 x^{3} + 1 = 0$

Subtract $1$ from both sides of the equation.

$2 x^{3} = - 1$

Divide each term in $2 x^{3} = - 1$ by $2$ and simplify.

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Divide each term in $2 x^{3} = - 1$ by $2$ .

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

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x^{3}$ by $1$ .

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

Simplify the right side.

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Move the negative in front of the fraction.

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

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

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

Simplify $\sqrt[3]{- \frac{1}{2}}$ .

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Rewrite $- \frac{1}{2}$ as ${({(- 1)}^{3})}^{3} \frac{1}{2}$ .

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Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \frac{1}{2}}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{1}{2}}$

Pull terms out from under the radical.

$x = {(- 1)}^{3} \sqrt[3]{\frac{1}{2}}$

Raise $- 1$ to the power of $3$ .

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

Rewrite $\sqrt[3]{\frac{1}{2}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{2}}$ .

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

Any root of $1$ is $1$ .

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

Multiply $\frac{1}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = - (\frac{1}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}})$

Combine and simplify the denominator.

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Multiply $\frac{1}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

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

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

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

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

$x = - \frac{{\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

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

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

Combine $\frac{1}{3}$ and $3$ .

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

Cancel the common factor of $3$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

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Rewrite ${\sqrt[3]{2}}^{2}$ as ${(2^{2})}^{\frac{1}{3}}$ .

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

Raise $2$ to the power of $2$ .

$x = - \frac{\sqrt[3]{4}}{2}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{\sqrt[3]{4}}{2}, 0)$

Find the y-intercepts.

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To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 2 {(0)}^{3} + 1$

Solve the equation.

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Remove parentheses.

$y = 2 \cdot 0^{3} + 1$

Remove parentheses.

$y = 2 {(0)}^{3} + 1$

Simplify $2 {(0)}^{3} + 1$ .

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Simplify each term.

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

$y = 2 \cdot 0 + 1$

Multiply $2$ by $0$ .

$y = 0 + 1$

Add $0$ and $1$ .

$y = 1$

y-intercept(s) in point form.

y-intercept(s): $(0, 1)$

List the intersections.

x-intercept(s): $(- \frac{\sqrt[3]{4}}{2}, 0)$

y-intercept(s): $(0, 1)$