Calculus Examples

$y = 4 x^{2}$ , $[0, \frac{7}{4}]$

Step 1

Substitute using the average rate of change formula.

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Step 1.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (\frac{7}{4}) - f (0)}{(\frac{7}{4}) - (0)}$

Step 1.2

Substitute the equation $y = 4 x^{2}$ for $f (\frac{7}{4})$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(4 {(\frac{7}{4})}^{2}) - (4 {(0)}^{2})}{(\frac{7}{4}) - (0)}$

Step 2

Simplify the expression.

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Step 2.1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 2.1.1

Multiply $\frac{4 {(\frac{7}{4})}^{2} - 1 \cdot 4 \cdot 0^{2}}{\frac{7}{4} - (0)}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{4 {(\frac{7}{4})}^{2} - 1 \cdot 4 \cdot 0^{2}}{\frac{7}{4} - (0)}$

Step 2.1.2

Combine.

$\frac{4 (4 {(\frac{7}{4})}^{2} - 1 \cdot 4 \cdot 0^{2})}{4 (\frac{7}{4} - (0))}$

Step 2.2

Apply the distributive property.

$\frac{4 (4 {(\frac{7}{4})}^{2}) + 4 (- 1 \cdot 4 \cdot 0^{2})}{4 (\frac{7}{4}) + 4 (- (0))}$

Step 2.3

Cancel the common factor of $4$ .

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Step 2.3.1

Cancel the common factor.

$\frac{4 (4 {(\frac{7}{4})}^{2}) + 4 (- 1 \cdot 4 \cdot 0^{2})}{4 (\frac{7}{4}) + 4 (- (0))}$

Step 2.3.2

Rewrite the expression.

$\frac{4 (4 {(\frac{7}{4})}^{2}) + 4 (- 1 \cdot 4 \cdot 0^{2})}{7 + 4 (- (0))}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Multiply $4$ by $4$ .

$\frac{16 {(\frac{7}{4})}^{2} + 4 (- 1 \cdot 4 \cdot 0^{2})}{7 + 4 (- (0))}$

Step 2.4.2

Apply the product rule to $\frac{7}{4}$ .

$\frac{16 \frac{7^{2}}{4^{2}} + 4 (- 1 \cdot 4 \cdot 0^{2})}{7 + 4 (- (0))}$

Step 2.4.3

Raise $7$ to the power of $2$ .

$\frac{16 \frac{49}{4^{2}} + 4 (- 1 \cdot 4 \cdot 0^{2})}{7 + 4 (- (0))}$

Step 2.4.4

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

$\frac{16 (\frac{49}{16}) + 4 (- 1 \cdot 4 \cdot 0^{2})}{7 + 4 (- (0))}$

Step 2.4.5

Cancel the common factor of $16$ .

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Step 2.4.5.1

Cancel the common factor.

$\frac{16 (\frac{49}{16}) + 4 (- 1 \cdot 4 \cdot 0^{2})}{7 + 4 (- (0))}$

Step 2.4.5.2

Rewrite the expression.

$\frac{49 + 4 (- 1 \cdot 4 \cdot 0^{2})}{7 + 4 (- (0))}$

Step 2.4.6

Multiply $- 1$ by $4$ .

$\frac{49 + 4 (- 4 \cdot 0^{2})}{7 + 4 (- (0))}$

Step 2.4.7

Raising $0$ to any positive power yields $0$ .

$\frac{49 + 4 (- 4 \cdot 0)}{7 + 4 (- (0))}$

Step 2.4.8

Multiply $4 (- 4 \cdot 0)$ .

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Step 2.4.8.1

Multiply $- 4$ by $0$ .

$\frac{49 + 4 \cdot 0}{7 + 4 (- (0))}$

Step 2.4.8.2

Multiply $4$ by $0$ .

$\frac{49 + 0}{7 + 4 (- (0))}$

Step 2.4.9

Add $49$ and $0$ .

$\frac{49}{7 + 4 (- (0))}$

Step 2.5

Simplify the denominator.

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Step 2.5.1

Multiply $4 (- (0))$ .

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Step 2.5.1.1

Multiply $- 1$ by $0$ .

$\frac{49}{7 + 4 \cdot 0}$

Step 2.5.1.2

Multiply $4$ by $0$ .

$\frac{49}{7 + 0}$

Step 2.5.2

Add $7$ and $0$ .

$\frac{49}{7}$

Step 2.6

Divide $49$ by $7$ .

$7$