Calculus Examples

$f (x) = x^{2} + 3 x + 34$ , $(- 3, 4)$

Step 1

Find the derivative of $f (x) = x^{2} + 3 x + 34$ .

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{2} + 3 x + 34$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$

Step 1.1.1.2

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

$2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$

Step 1.1.2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [34]$

Step 1.1.2.2

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

$2 x + 3 \cdot 1 + \frac{d}{d x} [34]$

Step 1.1.2.3

Multiply $3$ by $1$ .

$2 x + 3 + \frac{d}{d x} [34]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $34$ is constant with respect to $x$ , the derivative of $34$ with respect to $x$ is $0$ .

$2 x + 3 + 0$

Step 1.1.3.2

Add $2 x + 3$ and $0$ .

$f' (x) = 2 x + 3$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x + 3$ .

$2 x + 3$

Step 2

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 3

$f' (x)$ is continuous on $[- 3, 4]$ .

$f' (x)$ is continuous

Step 4

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 5

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

$A (x) = \frac{1}{4 + 3} (\int_{- 3}^{4} 2 x + 3 d x)$

Step 6

Split the single integral into multiple integrals.

$A (x) = \frac{1}{4 + 3} (\int_{- 3}^{4} 2 x d x + \int_{- 3}^{4} 3 d x)$

Step 7

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

$A (x) = \frac{1}{4 + 3} (2 \int_{- 3}^{4} x d x + \int_{- 3}^{4} 3 d x)$

Step 8

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

$A (x) = \frac{1}{4 + 3} (2 (\frac{1}{2} x^{2}]_{- 3}^{4}) + \int_{- 3}^{4} 3 d x)$

Step 9

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

$A (x) = \frac{1}{4 + 3} (2 (\frac{x^{2}}{2}]_{- 3}^{4}) + \int_{- 3}^{4} 3 d x)$

Step 10

Apply the constant rule.

$A (x) = \frac{1}{4 + 3} (2 (\frac{x^{2}}{2}]_{- 3}^{4}) + 3 x]_{- 3}^{4})$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $- 3$ .

$A (x) = \frac{1}{4 + 3} (2 ((\frac{4^{2}}{2}) - \frac{{(- 3)}^{2}}{2}) + 3 x]_{- 3}^{4})$

Step 11.2

Evaluate $3 x$ at $4$ and at $- 3$ .

$A (x) = \frac{1}{4 + 3} (2 (\frac{4^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3

Simplify.

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Step 11.3.1

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

$A (x) = \frac{1}{4 + 3} (2 (\frac{16}{2} - \frac{{(- 3)}^{2}}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.2

Cancel the common factor of $16$ and $2$ .

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Step 11.3.2.1

Factor $2$ out of $16$ .

$A (x) = \frac{1}{4 + 3} (2 (\frac{2 \cdot 8}{2} - \frac{{(- 3)}^{2}}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.2.2

Cancel the common factors.

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Step 11.3.2.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{4 + 3} (2 (\frac{2 \cdot 8}{2 (1)} - \frac{{(- 3)}^{2}}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 + 3} (2 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{{(- 3)}^{2}}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 + 3} (2 (\frac{8}{1} - \frac{{(- 3)}^{2}}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.2.2.4

Divide $8$ by $1$ .

$A (x) = \frac{1}{4 + 3} (2 (8 - \frac{{(- 3)}^{2}}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.3

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

$A (x) = \frac{1}{4 + 3} (2 (8 - \frac{9}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.4

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{4 + 3} (2 (8 \cdot \frac{2}{2} - \frac{9}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.5

Combine $8$ and $\frac{2}{2}$ .

$A (x) = \frac{1}{4 + 3} (2 (\frac{8 \cdot 2}{2} - \frac{9}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.6

Combine the numerators over the common denominator.

$A (x) = \frac{1}{4 + 3} (2 (\frac{8 \cdot 2 - 9}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.7

Simplify the numerator.

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Step 11.3.7.1

Multiply $8$ by $2$ .

$A (x) = \frac{1}{4 + 3} (2 (\frac{16 - 9}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.7.2

Subtract $9$ from $16$ .

$A (x) = \frac{1}{4 + 3} (2 (\frac{7}{2}) + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.8

Combine $2$ and $\frac{7}{2}$ .

$A (x) = \frac{1}{4 + 3} (\frac{2 \cdot 7}{2} + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.9

Multiply $2$ by $7$ .

$A (x) = \frac{1}{4 + 3} (\frac{14}{2} + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.10

Cancel the common factor of $14$ and $2$ .

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Step 11.3.10.1

Factor $2$ out of $14$ .

$A (x) = \frac{1}{4 + 3} (\frac{2 \cdot 7}{2} + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.10.2

Cancel the common factors.

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Step 11.3.10.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{4 + 3} (\frac{2 \cdot 7}{2 (1)} + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.10.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 + 3} (\frac{2 \cdot 7}{2 \cdot 1} + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.10.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 + 3} (\frac{7}{1} + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.10.2.4

Divide $7$ by $1$ .

$A (x) = \frac{1}{4 + 3} (7 + 3 \cdot 4 - 3 \cdot - 3)$

Step 11.3.11

Multiply $3$ by $4$ .

$A (x) = \frac{1}{4 + 3} (7 + 12 - 3 \cdot - 3)$

Step 11.3.12

Multiply $- 3$ by $- 3$ .

$A (x) = \frac{1}{4 + 3} (7 + 12 + 9)$

Step 11.3.13

Add $12$ and $9$ .

$A (x) = \frac{1}{4 + 3} (7 + 21)$

Step 11.3.14

Add $7$ and $21$ .

$A (x) = \frac{1}{4 + 3} (28)$

Step 12

Add $4$ and $3$ .

$A (x) = \frac{1}{7} \cdot 28$

Step 13

Cancel the common factor of $7$ .

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Step 13.1

Factor $7$ out of $28$ .

$A (x) = \frac{1}{7} \cdot (7 (4))$

Step 13.2

Cancel the common factor.

$A (x) = \frac{1}{7} \cdot (7 \cdot 4)$

Step 13.3

Rewrite the expression.

$A (x) = 4$

Step 14

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