Calculus Examples

$s (t) = 2 t^{2} + 6 t$ , $t = 2$

Step 1

The percentage rate of change for the function is the value of the derivative (rate of change) at $2$ over the value of the function at $2$ .

$\frac{f' (2)}{f (2)}$

Step 2

Substitute the functions into the formula to find the function for the percentage rate of change.

$\frac{4 t + 6}{2 t^{2} + 6 t}$

Step 3

Factor $2$ out of $4 t + 6$ .

Tap for more steps...

Step 3.1

Factor $2$ out of $4 t$ .

$\frac{f' (t)}{f (t)} = \frac{2 (2 t) + 6}{2 t^{2} + 6 t}$

Step 3.2

Factor $2$ out of $6$ .

$\frac{f' (t)}{f (t)} = \frac{2 (2 t) + 2 (3)}{2 t^{2} + 6 t}$

Step 3.3

Factor $2$ out of $2 (2 t) + 2 (3)$ .

$\frac{f' (t)}{f (t)} = \frac{2 (2 t + 3)}{2 t^{2} + 6 t}$

Step 4

Factor $2 t$ out of $2 t^{2} + 6 t$ .

Tap for more steps...

Step 4.1

Factor $2 t$ out of $2 t^{2}$ .

$\frac{f' (t)}{f (t)} = \frac{2 (2 t + 3)}{2 t (t) + 6 t}$

Step 4.2

Factor $2 t$ out of $6 t$ .

$\frac{f' (t)}{f (t)} = \frac{2 (2 t + 3)}{2 t (t) + 2 t (3)}$

Step 4.3

Factor $2 t$ out of $2 t (t) + 2 t (3)$ .

$\frac{f' (t)}{f (t)} = \frac{2 (2 t + 3)}{2 t (t + 3)}$

Step 5

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.1

Cancel the common factor.

$\frac{f' (t)}{f (t)} = \frac{2 (2 t + 3)}{2 t (t + 3)}$

Step 5.2

Rewrite the expression.

$\frac{f' (t)}{f (t)} = \frac{2 t + 3}{t (t + 3)}$

Step 6

Evaluate the formula at $t = 2$ .

$\frac{2 (2) + 3}{(2) ((2) + 3)}$

Step 7

Simplify $\frac{2 (2) + 3}{(2) ((2) + 3)}$ .

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Multiply $2$ by $2$ .

$\frac{4 + 3}{2 (2 + 3)}$

Step 7.1.2

Add $4$ and $3$ .

$\frac{7}{2 (2 + 3)}$

Step 7.2

Simplify the expression.

Tap for more steps...

Step 7.2.1

Add $2$ and $3$ .

$\frac{7}{2 \cdot 5}$

Step 7.2.2

Multiply $2$ by $5$ .

$\frac{7}{10}$

Step 8

Convert $\frac{7}{10}$ to a decimal.

$0.7$

Step 9

Convert $0.7$ to a percentage.

$70$ %