Calculus Examples

$x^{2} \ln (x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x = 0$

Step 1.2

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = {(1)}^{2} \ln (1)$

Step 2.2

Simplify the result.

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Step 2.2.1

One to any power is one.

$f (1) = 1 \ln (1)$

Step 2.2.2

Multiply $\ln (1)$ by $1$ .

$f (1) = \ln (1)$

Step 2.2.3

The natural logarithm of $1$ is $0$ .

$f (1) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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Step 3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = {(2)}^{2} \ln (2)$

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (2) = 4 \ln (2)$

Step 3.2.2

Simplify $4 \ln (2)$ by moving $4$ inside the logarithm.

$f (2) = \ln (2^{4})$

Step 3.2.3

Raise $2$ to the power of $4$ .

$f (2) = \ln (16)$

Step 3.2.4

The final answer is $\ln (16)$ .

$\ln (16)$

Step 3.3

Convert $\ln (16)$ to decimal.

$y = 2.77258872$

Step 4

Find the point at $x = 3$ .

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Step 4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = {(3)}^{2} \ln (3)$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $3$ to the power of $2$ .

$f (3) = 9 \ln (3)$

Step 4.2.2

Simplify $9 \ln (3)$ by moving $9$ inside the logarithm.

$f (3) = \ln (3^{9})$

Step 4.2.3

Raise $3$ to the power of $9$ .

$f (3) = \ln (19683)$

Step 4.2.4

The final answer is $\ln (19683)$ .

$\ln (19683)$

Step 4.3

Convert $\ln (19683)$ to decimal.

$y = 9.88751059$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 2.77258872), (3, 9.88751059)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 2.773 \\ 3 & 9.888 \end{array}$

Step 6