Calculus Examples

$y = 4 x^{2} + 3$ , $(1, 7)$ , $(1.1, 7.84)$

Step 1

Write $y = 4 x^{2} + 3$ as a function.

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

Step 2

Substitute using the average rate of change formula.

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Step 2.1

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.

$\frac{f (1.1) - f (1)}{(1.1) - (1)}$

Step 2.2

Substitute the equation $y = 4 x^{2} + 3$ for $f (1.1)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(4 {(1.1)}^{2} + 3) - (4 {(1)}^{2} + 3)}{(1.1) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $1.1$ to the power of $2$ .

$\frac{4 \cdot 1.21 + 3 - (4 \cdot 1^{2} + 3)}{1.1 - (1)}$

Step 3.1.2

Multiply $4$ by $1.21$ .

$\frac{4.84 + 3 - (4 \cdot 1^{2} + 3)}{1.1 - (1)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

One to any power is one.

$\frac{4.84 + 3 - (4 \cdot 1 + 3)}{1.1 - (1)}$

Step 3.1.3.2

Multiply $4$ by $1$ .

$\frac{4.84 + 3 - (4 + 3)}{1.1 - (1)}$

Step 3.1.4

Add $4$ and $3$ .

$\frac{4.84 + 3 - 1 \cdot 7}{1.1 - (1)}$

Step 3.1.5

Multiply $- 1$ by $7$ .

$\frac{4.84 + 3 - 7}{1.1 - (1)}$

Step 3.1.6

Add $4.84$ and $3$ .

$\frac{7.84 - 7}{1.1 - (1)}$

Step 3.1.7

Subtract $7$ from $7.84$ .

$\frac{0.84}{1.1 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{0.84}{1.1 - 1}$

Step 3.2.2

Subtract $1$ from $1.1$ .

$\frac{0.84}{0.1}$

Step 3.3

Divide $0.84$ by $0.1$ .

$8.4$