Calculus Examples

$p (x) = \ln (- x^{2} + 3 x^{2} + 72 x + 1)$ , $0 < x < 10$

Step 1

To find the average value of a function, the function should be continuous on the closed interval $[a, b]$ . To find whether $p (x)$ is continuous on $[0, 10]$ or not, find the domain of $p (x) = \ln (- x^{2} + 3 x^{2} + 72 x + 1)$ .

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Step 1.1

Set the argument in $\ln (- x^{2} + 3 x^{2} + 72 x + 1)$ greater than $0$ to find where the expression is defined.

$- x^{2} + 3 x^{2} + 72 x + 1 > 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $- x^{2}$ and $3 x^{2}$ .

$2 x^{2} + 72 x + 1 > 0$

Step 1.2.2

Convert the inequality to an equation.

$2 x^{2} + 72 x + 1 = 0$

Step 1.2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.4

Substitute the values $a = 2$ , $b = 72$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 72 \pm \sqrt{72^{2} - 4 \cdot (2 \cdot 1)}}{2 \cdot 2}$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

Raise $72$ to the power of $2$ .

$x = \frac{- 72 \pm \sqrt{5184 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 72 \pm \sqrt{5184 - 8 \cdot 1}}{2 \cdot 2}$

Step 1.2.5.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{- 72 \pm \sqrt{5184 - 8}}{2 \cdot 2}$

Step 1.2.5.1.3

Subtract $8$ from $5184$ .

$x = \frac{- 72 \pm \sqrt{5176}}{2 \cdot 2}$

Step 1.2.5.1.4

Rewrite $5176$ as $2^{2} \cdot 1294$ .

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Step 1.2.5.1.4.1

Factor $4$ out of $5176$ .

$x = \frac{- 72 \pm \sqrt{4 (1294)}}{2 \cdot 2}$

Step 1.2.5.1.4.2

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

$x = \frac{- 72 \pm \sqrt{2^{2} \cdot 1294}}{2 \cdot 2}$

Step 1.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 72 \pm 2 \sqrt{1294}}{2 \cdot 2}$

Step 1.2.5.2

Multiply $2$ by $2$ .

$x = \frac{- 72 \pm 2 \sqrt{1294}}{4}$

Step 1.2.5.3

Simplify $\frac{- 72 \pm 2 \sqrt{1294}}{4}$ .

$x = \frac{- 36 \pm \sqrt{1294}}{2}$

Step 1.2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $72$ to the power of $2$ .

$x = \frac{- 72 \pm \sqrt{5184 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 72 \pm \sqrt{5184 - 8 \cdot 1}}{2 \cdot 2}$

Step 1.2.6.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{- 72 \pm \sqrt{5184 - 8}}{2 \cdot 2}$

Step 1.2.6.1.3

Subtract $8$ from $5184$ .

$x = \frac{- 72 \pm \sqrt{5176}}{2 \cdot 2}$

Step 1.2.6.1.4

Rewrite $5176$ as $2^{2} \cdot 1294$ .

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Step 1.2.6.1.4.1

Factor $4$ out of $5176$ .

$x = \frac{- 72 \pm \sqrt{4 (1294)}}{2 \cdot 2}$

Step 1.2.6.1.4.2

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

$x = \frac{- 72 \pm \sqrt{2^{2} \cdot 1294}}{2 \cdot 2}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 72 \pm 2 \sqrt{1294}}{2 \cdot 2}$

Step 1.2.6.2

Multiply $2$ by $2$ .

$x = \frac{- 72 \pm 2 \sqrt{1294}}{4}$

Step 1.2.6.3

Simplify $\frac{- 72 \pm 2 \sqrt{1294}}{4}$ .

$x = \frac{- 36 \pm \sqrt{1294}}{2}$

Step 1.2.6.4

Change the $\pm$ to $+$ .

$x = \frac{- 36 + \sqrt{1294}}{2}$

Step 1.2.6.5

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{- 1 \cdot 36 + \sqrt{1294}}{2}$

Step 1.2.6.6

Factor $- 1$ out of $\sqrt{1294}$ .

$x = \frac{- 1 \cdot 36 - 1 (- \sqrt{1294})}{2}$

Step 1.2.6.7

Factor $- 1$ out of $- 1 (36) - 1 (- \sqrt{1294})$ .

$x = \frac{- 1 (36 - \sqrt{1294})}{2}$

Step 1.2.6.8

Move the negative in front of the fraction.

$x = - \frac{36 - \sqrt{1294}}{2}$

Step 1.2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

Raise $72$ to the power of $2$ .

$x = \frac{- 72 \pm \sqrt{5184 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 72 \pm \sqrt{5184 - 8 \cdot 1}}{2 \cdot 2}$

Step 1.2.7.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{- 72 \pm \sqrt{5184 - 8}}{2 \cdot 2}$

Step 1.2.7.1.3

Subtract $8$ from $5184$ .

$x = \frac{- 72 \pm \sqrt{5176}}{2 \cdot 2}$

Step 1.2.7.1.4

Rewrite $5176$ as $2^{2} \cdot 1294$ .

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Step 1.2.7.1.4.1

Factor $4$ out of $5176$ .

$x = \frac{- 72 \pm \sqrt{4 (1294)}}{2 \cdot 2}$

Step 1.2.7.1.4.2

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

$x = \frac{- 72 \pm \sqrt{2^{2} \cdot 1294}}{2 \cdot 2}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 72 \pm 2 \sqrt{1294}}{2 \cdot 2}$

Step 1.2.7.2

Multiply $2$ by $2$ .

$x = \frac{- 72 \pm 2 \sqrt{1294}}{4}$

Step 1.2.7.3

Simplify $\frac{- 72 \pm 2 \sqrt{1294}}{4}$ .

$x = \frac{- 36 \pm \sqrt{1294}}{2}$

Step 1.2.7.4

Change the $\pm$ to $-$ .

$x = \frac{- 36 - \sqrt{1294}}{2}$

Step 1.2.7.5

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{- 1 \cdot 36 - \sqrt{1294}}{2}$

Step 1.2.7.6

Factor $- 1$ out of $- \sqrt{1294}$ .

$x = \frac{- 1 \cdot 36 - (\sqrt{1294})}{2}$

Step 1.2.7.7

Factor $- 1$ out of $- 1 (36) - (\sqrt{1294})$ .

$x = \frac{- 1 (36 + \sqrt{1294})}{2}$

Step 1.2.7.8

Move the negative in front of the fraction.

$x = - \frac{36 + \sqrt{1294}}{2}$

Step 1.2.8

Consolidate the solutions.

$x = - \frac{36 - \sqrt{1294}}{2}, - \frac{36 + \sqrt{1294}}{2}$

Step 1.2.9

Use each root to create test intervals.

$x < - \frac{36 + \sqrt{1294}}{2}$

$- \frac{36 + \sqrt{1294}}{2} < x < - \frac{36 - \sqrt{1294}}{2}$

$x > - \frac{36 - \sqrt{1294}}{2}$

Step 1.2.10

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 1.2.10.1

Test a value on the interval $x < - \frac{36 + \sqrt{1294}}{2}$ to see if it makes the inequality true.

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Step 1.2.10.1.1

Choose a value on the interval $x < - \frac{36 + \sqrt{1294}}{2}$ and see if this value makes the original inequality true.

$x = - 38$

Step 1.2.10.1.2

Replace $x$ with $- 38$ in the original inequality.

$- {(- 38)}^{2} + 3 {(- 38)}^{2} + 72 (- 38) + 1 > 0$

Step 1.2.10.1.3

The left side $153$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.10.2

Test a value on the interval $- \frac{36 + \sqrt{1294}}{2} < x < - \frac{36 - \sqrt{1294}}{2}$ to see if it makes the inequality true.

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Step 1.2.10.2.1

Choose a value on the interval $- \frac{36 + \sqrt{1294}}{2} < x < - \frac{36 - \sqrt{1294}}{2}$ and see if this value makes the original inequality true.

$x = - 3$

Step 1.2.10.2.2

Replace $x$ with $- 3$ in the original inequality.

$- {(- 3)}^{2} + 3 {(- 3)}^{2} + 72 (- 3) + 1 > 0$

Step 1.2.10.2.3

The left side $- 197$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 1.2.10.3

Test a value on the interval $x > - \frac{36 - \sqrt{1294}}{2}$ to see if it makes the inequality true.

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Step 1.2.10.3.1

Choose a value on the interval $x > - \frac{36 - \sqrt{1294}}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 1.2.10.3.2

Replace $x$ with $0$ in the original inequality.

$- {(0)}^{2} + 3 {(0)}^{2} + 72 (0) + 1 > 0$

Step 1.2.10.3.3

The left side $1$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.10.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - \frac{36 + \sqrt{1294}}{2}$ True

$- \frac{36 + \sqrt{1294}}{2} < x < - \frac{36 - \sqrt{1294}}{2}$ False

$x > - \frac{36 - \sqrt{1294}}{2}$ True

$x < - \frac{36 + \sqrt{1294}}{2}$ True

$- \frac{36 + \sqrt{1294}}{2} < x < - \frac{36 - \sqrt{1294}}{2}$ False

$x > - \frac{36 - \sqrt{1294}}{2}$ True

Step 1.2.11

The solution consists of all of the true intervals.

$x < - \frac{36 + \sqrt{1294}}{2}$ or $x > - \frac{36 - \sqrt{1294}}{2}$

Step 1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - \frac{36 + \sqrt{1294}}{2}) \cup (- \frac{36 - \sqrt{1294}}{2}, \infty)$

Set-Builder Notation:

${x | x < - \frac{36 + \sqrt{1294}}{2}, x > - \frac{36 - \sqrt{1294}}{2}}$

Interval Notation:

$(- \infty, - \frac{36 + \sqrt{1294}}{2}) \cup (- \frac{36 - \sqrt{1294}}{2}, \infty)$

Set-Builder Notation:

${x | x < - \frac{36 + \sqrt{1294}}{2}, x > - \frac{36 - \sqrt{1294}}{2}}$

Step 2

$p (x)$ is continuous on $[0, 10]$ .

$p (x)$ is continuous

Step 3

The average value of function $p$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 4

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{10 - 0} (\int_{0}^{10} \ln (- x^{2} + 3 x^{2} + 72 x + 1) d x)$

Step 5

Simplify the denominator.

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Step 5.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{10 + 0} \cdot \int_{0}^{10} \ln (- x^{2} + 3 x^{2} + 72 x + 1) d x$

Step 5.2

Add $10$ and $0$ .

$A (x) = \frac{1}{10} \cdot \int_{0}^{10} \ln (- x^{2} + 3 x^{2} + 72 x + 1) d x$

Step 6

Simplify terms.

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Step 6.1

Add $- x^{2}$ and $3 x^{2}$ .

$A (x) = \frac{1}{10} \cdot \int_{0}^{10} \ln (2 x^{2} + 72 x + 1) d x$

Step 6.2

Combine $\frac{1}{10}$ and $\int_{0}^{10} \ln (2 x^{2} + 72 x + 1) d x$ .

$A (x) = \frac{\int_{0}^{10} \ln (2 x^{2} + 72 x + 1) d x}{10}$

Step 7