Calculus Examples

y = 25 - x^{2}

$y = 25 - x^{2}$ ,

[- 5, 5]

$[- 5, 5]$

Step 1

Write

y = 25 - x^{2}

$y = 25 - x^{2}$ as a function.

f (x) = 25 - x^{2}

$f (x) = 25 - x^{2}$

Step 2

Find the derivative of

f (x) = 25 - x^{2}

$f (x) = 25 - x^{2}$ .

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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

25 - x^{2}

$25 - x^{2}$ with respect to

x

$x$ is

\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]

$\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]$ .

\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]

$\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]$

Step 2.1.1.2

Since

25

$25$ is constant with respect to

x

$x$ , the derivative of

25

$25$ with respect to

x

$x$ is

0

$0$ .

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

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

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

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

Step 2.1.2

Evaluate

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

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

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Step 2.1.2.1

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- x^{2}

$- x^{2}$ with respect to

x

$x$ is

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

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

0 - \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}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

0 - (2 x)

$0 - (2 x)$

Step 2.1.2.3

Multiply

2

$2$ by

- 1

$- 1$ .

0 - 2 x

$0 - 2 x$

0 - 2 x

$0 - 2 x$

Step 2.1.3

Subtract

2 x

$2 x$ from

0

$0$ .

$f' (x) = - 2 x$

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

$[- 5, 5]$ .

$f' (x)$ 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.

$A (x) = \frac{1}{5 + 5} (\int_{- 5}^{5} - 2 x d x)$

Step 7

Since

$- 2$ is constant with respect to

$x$ , move

$- 2$ out of the integral.

$A (x) = \frac{1}{5 + 5} (- 2 \int_{- 5}^{5} x d x)$

Step 8

By the Power Rule, the integral of

$x$ with respect to

$x$ is

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

$A (x) = \frac{1}{5 + 5} (- 2 (\frac{1}{2} x^{2}]_{- 5}^{5}))$

Step 9

Simplify the answer.

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Step 9.1

Combine

$\frac{1}{2}$ and

$x^{2}$ .

$A (x) = \frac{1}{5 + 5} (- 2 (\frac{x^{2}}{2}]_{- 5}^{5}))$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate

$\frac{x^{2}}{2}$ at

$5$ and at

$- 5$ .

$A (x) = \frac{1}{5 + 5} (- 2 ((\frac{5^{2}}{2}) - \frac{{(- 5)}^{2}}{2}))$

Step 9.2.2

Simplify.

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Step 9.2.2.1

Raise

$5$ to the power of

$2$ .

$A (x) = \frac{1}{5 + 5} (- 2 (\frac{25}{2} - \frac{{(- 5)}^{2}}{2}))$

Step 9.2.2.2

Raise

$- 5$ to the power of

$2$ .

$A (x) = \frac{1}{5 + 5} (- 2 (\frac{25}{2} - \frac{25}{2}))$

Step 9.2.2.3

Combine the numerators over the common denominator.

$A (x) = \frac{1}{5 + 5} (- 2 \frac{25 - 25}{2})$

Step 9.2.2.4

Subtract

$25$ from

$25$ .

$A (x) = \frac{1}{5 + 5} (- 2 (\frac{0}{2}))$

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

$0$ .

$A (x) = \frac{1}{5 + 5} (- 2 \frac{2 (0)}{2})$

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

$2$ .

$A (x) = \frac{1}{5 + 5} (- 2 \frac{2 \cdot 0}{2 \cdot 1})$

Step 9.2.2.5.2.2

Cancel the common factor.

$A (x) = \frac{1}{5 + 5} (- 2 \frac{2 \cdot 0}{2 \cdot 1})$

Step 9.2.2.5.2.3

Rewrite the expression.

$A (x) = \frac{1}{5 + 5} (- 2 (\frac{0}{1}))$

Step 9.2.2.5.2.4

Divide

$0$ by

$1$ .

$A (x) = \frac{1}{5 + 5} (- 2 \cdot 0)$

Step 9.2.2.6

Multiply

$- 2$ by

$0$ .

$A (x) = \frac{1}{5 + 5} (0)$

Step 10

Add

$5$ and

$5$ .

$A (x) = \frac{1}{10} \cdot 0$

Step 11

Multiply

$\frac{1}{10}$ by

$0$ .

$A (x) = 0$

Step 12