Calculus Examples

$4 \sum_{t = 0}^{3} - 2 t + 4 t^{2}$

Step 1

Expand the series for each value of $t$ .

$4 (- 2 \cdot 0 + 4 \cdot 0^{2} - 2 \cdot 1 + 4 \cdot 1^{2} - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2

Simplify.

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Step 2.1

Multiply $- 2$ by $0$ .

$4 (0 + 4 \cdot 0^{2} - 2 \cdot 1 + 4 \cdot 1^{2} - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.2

Raise $0$ to the power of $2$ .

$4 (0 + 4 \cdot 0 - 2 \cdot 1 + 4 \cdot 1^{2} - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.3

Multiply $4$ by $0$ .

$4 (0 + 0 - 2 \cdot 1 + 4 \cdot 1^{2} - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.4

Add $0$ and $0$ .

$4 (0 - 2 \cdot 1 + 4 \cdot 1^{2} - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.5

Multiply $- 2$ by $1$ .

$4 (0 - 2 + 4 \cdot 1^{2} - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.6

Raise $1$ to the power of $2$ .

$4 (0 - 2 + 4 \cdot 1 - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.7

Multiply $4$ by $1$ .

$4 (0 - 2 + 4 - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.8

Add $- 2$ and $4$ .

$4 (0 + 2 - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.9

Add $0$ and $2$ .

$4 (2 - 2 \cdot 2 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.10

Multiply $- 2$ by $2$ .

$4 (2 - 4 + 4 \cdot 2^{2} - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.11

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

$4 (2 - 4 + 4 \cdot 4 - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.12

Multiply $4$ by $4$ .

$4 (2 - 4 + 16 - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.13

Add $- 4$ and $16$ .

$4 (2 + 12 - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.14

Add $2$ and $12$ .

$4 (14 - 2 \cdot 3 + 4 \cdot 3^{2})$

Step 2.15

Multiply $- 2$ by $3$ .

$4 (14 - 6 + 4 \cdot 3^{2})$

Step 2.16

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

$4 (14 - 6 + 4 \cdot 9)$

Step 2.17

Multiply $4$ by $9$ .

$4 (14 - 6 + 36)$

Step 2.18

Add $- 6$ and $36$ .

$4 (14 + 30)$

Step 2.19

Add $14$ and $30$ .

$4 \cdot 44$

Step 2.20

Multiply $4$ by $44$ .

$176$