Calculus Examples

$y = 81 - x^{2}$ , $[- 9, 9]$

Step 1

Write $y = 81 - x^{2}$ as a function.

$f (x) = 81 - x^{2}$

Step 2

Find the derivative of $f (x) = 81 - x^{2}$ .

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate.

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Step 2.1.1.1

By the Sum Rule, the derivative of $81 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [81] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [81] + \frac{d}{d x} [- x^{2}]$

Step 2.1.1.2

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 2.1.2

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

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Step 2.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$0 - \frac{d}{d x} [x^{2}]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (2 x)$

Step 2.1.2.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 2.1.3

Subtract $2 x$ from $0$ .

$f' (x) = - 2 x$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $- 2 x$ .

$- 2 x$

Step 3

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

$f' (x)$ is continuous on $[- 9, 9]$ .

$f' (x)$ is continuous

Step 5

The average value of function $f'$ 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 6

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

$A (x) = \frac{1}{9 + 9} (\int_{- 9}^{9} - 2 x d x)$

Step 7

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$A (x) = \frac{1}{9 + 9} (- 2 \int_{- 9}^{9} x d x)$

Step 8

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$A (x) = \frac{1}{9 + 9} (- 2 (\frac{1}{2} x^{2}]_{- 9}^{9}))$

Step 9

Simplify the answer.

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Step 9.1

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

$A (x) = \frac{1}{9 + 9} (- 2 (\frac{x^{2}}{2}]_{- 9}^{9}))$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{x^{2}}{2}$ at $9$ and at $- 9$ .

$A (x) = \frac{1}{9 + 9} (- 2 ((\frac{9^{2}}{2}) - \frac{{(- 9)}^{2}}{2}))$

Step 9.2.2

Simplify.

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Step 9.2.2.1

Raise $9$ to the power of $2$ .

$A (x) = \frac{1}{9 + 9} (- 2 (\frac{81}{2} - \frac{{(- 9)}^{2}}{2}))$

Step 9.2.2.2

Raise $- 9$ to the power of $2$ .

$A (x) = \frac{1}{9 + 9} (- 2 (\frac{81}{2} - \frac{81}{2}))$

Step 9.2.2.3

Combine the numerators over the common denominator.

$A (x) = \frac{1}{9 + 9} (- 2 \frac{81 - 81}{2})$

Step 9.2.2.4

Subtract $81$ from $81$ .

$A (x) = \frac{1}{9 + 9} (- 2 (\frac{0}{2}))$

Step 9.2.2.5

Cancel the common factor of $0$ and $2$ .

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Step 9.2.2.5.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{9 + 9} (- 2 \frac{2 (0)}{2})$

Step 9.2.2.5.2

Cancel the common factors.

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Step 9.2.2.5.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{9 + 9} (- 2 \frac{2 \cdot 0}{2 \cdot 1})$

Step 9.2.2.5.2.2

Cancel the common factor.

$A (x) = \frac{1}{9 + 9} (- 2 \frac{2 \cdot 0}{2 \cdot 1})$

Step 9.2.2.5.2.3

Rewrite the expression.

$A (x) = \frac{1}{9 + 9} (- 2 (\frac{0}{1}))$

Step 9.2.2.5.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{9 + 9} (- 2 \cdot 0)$

Step 9.2.2.6

Multiply $- 2$ by $0$ .

$A (x) = \frac{1}{9 + 9} (0)$

Step 10

Add $9$ and $9$ .

$A (x) = \frac{1}{18} \cdot 0$

Step 11

Multiply $\frac{1}{18}$ by $0$ .

$A (x) = 0$

Step 12