Calculus Examples

$1.4 \sum_{k = 1}^{10} 49 - {(- 7 + 1.4 k)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(- 7 + 1.4 k)}^{2}$ as $(- 7 + 1.4 k) (- 7 + 1.4 k)$ .

$49 - ((- 7 + 1.4 k) (- 7 + 1.4 k))$

Step 1.1.2

Expand $(- 7 + 1.4 k) (- 7 + 1.4 k)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$49 - (- 7 (- 7 + 1.4 k) + 1.4 k (- 7 + 1.4 k))$

Step 1.1.2.2

Apply the distributive property.

$49 - (- 7 \cdot - 7 - 7 (1.4 k) + 1.4 k (- 7 + 1.4 k))$

Step 1.1.2.3

Apply the distributive property.

$49 - (- 7 \cdot - 7 - 7 (1.4 k) + 1.4 k \cdot - 7 + 1.4 k (1.4 k))$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $- 7$ by $- 7$ .

$49 - (49 - 7 (1.4 k) + 1.4 k \cdot - 7 + 1.4 k (1.4 k))$

Step 1.1.3.1.2

Multiply $1.4$ by $- 7$ .

$49 - (49 - 9.8 k + 1.4 k \cdot - 7 + 1.4 k (1.4 k))$

Step 1.1.3.1.3

Multiply $- 7$ by $1.4$ .

$49 - (49 - 9.8 k - 9.8 k + 1.4 k (1.4 k))$

Step 1.1.3.1.4

Rewrite using the commutative property of multiplication.

$49 - (49 - 9.8 k - 9.8 k + 1.4 \cdot 1.4 k \cdot k)$

Step 1.1.3.1.5

Multiply $k$ by $k$ by adding the exponents.

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Step 1.1.3.1.5.1

Move $k$ .

$49 - (49 - 9.8 k - 9.8 k + 1.4 \cdot 1.4 (k \cdot k))$

Step 1.1.3.1.5.2

Multiply $k$ by $k$ .

$49 - (49 - 9.8 k - 9.8 k + 1.4 \cdot 1.4 k^{2})$

Step 1.1.3.1.6

Multiply $1.4$ by $1.4$ .

$49 - (49 - 9.8 k - 9.8 k + 1.96 k^{2})$

Step 1.1.3.2

Subtract $9.8 k$ from $- 9.8 k$ .

$49 - (49 - 19.6 k + 1.96 k^{2})$

Step 1.1.4

Apply the distributive property.

$49 - 1 \cdot 49 - (- 19.6 k) - (1.96 k^{2})$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Multiply $- 1$ by $49$ .

$49 - 49 - (- 19.6 k) - (1.96 k^{2})$

Step 1.1.5.2

Multiply $- 19.6$ by $- 1$ .

$49 - 49 + 19.6 k - (1.96 k^{2})$

Step 1.1.5.3

Multiply $1.96$ by $- 1$ .

$49 - 49 + 19.6 k - 1.96 k^{2}$

Step 1.2

Subtract $49$ from $49$ .

$0 + 19.6 k - 1.96 k^{2}$

Step 1.3

Add $0$ and $19.6 k$ .

$19.6 k - 1.96 k^{2}$

Step 1.4

Rewrite the summation.

$1.4 \sum_{k = 1}^{10} 19.6 k - 1.96 k^{2}$

Step 2

Split the summation into smaller summations that fit the summation rules.

$1.4 \sum_{k = 1}^{10} 19.6 k - 1.96 k^{2} = (1.4 \cdot 19.6) \sum_{k = 1}^{10} k + (1.4 \cdot - 1.96) \sum_{k = 1}^{10} k^{2}$

Step 3

Evaluate $(1.4 \cdot 19.6) \sum_{k = 1}^{10} k$ .

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Step 3.1

The formula for the summation of a polynomial with degree $1$ is:

$\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}$

Step 3.2

Substitute the values into the formula and make sure to multiply by the front term.

$(1.4 \cdot 19.6) (\frac{10 (10 + 1)}{2})$

Step 3.3

Simplify.

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Step 3.3.1

Add $10$ and $1$ .

$1.4 \cdot 19.6 \frac{10 \cdot 11}{2}$

Step 3.3.2

Multiply $10$ by $11$ .

$1.4 \cdot 19.6 \frac{110}{2}$

Step 3.3.3

Multiply $1.4$ by $19.6$ .

$27.44 (\frac{110}{2})$

Step 3.3.4

Divide $110$ by $2$ .

$27.44 \cdot 55$

Step 3.3.5

Multiply $27.44$ by $55$ .

$1509.2$

Step 4

Evaluate $(1.4 \cdot - 1.96) \sum_{k = 1}^{10} k^{2}$ .

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Step 4.1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 4.2

Substitute the values into the formula and make sure to multiply by the front term.

$(1.4 \cdot - 1.96) (\frac{10 (10 + 1) (2 \cdot 10 + 1)}{6})$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Add $10$ and $1$ .

$1.4 \cdot - 1.96 \frac{10 \cdot 11 (2 \cdot 10 + 1)}{6}$

Step 4.3.1.2

Combine exponents.

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Step 4.3.1.2.1

Multiply $10$ by $11$ .

$1.4 \cdot - 1.96 \frac{110 (2 \cdot 10 + 1)}{6}$

Step 4.3.1.2.2

Multiply $2$ by $10$ .

$1.4 \cdot - 1.96 \frac{110 (20 + 1)}{6}$

Step 4.3.1.3

Add $20$ and $1$ .

$1.4 \cdot - 1.96 \frac{110 \cdot 21}{6}$

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Multiply $110$ by $21$ .

$1.4 \cdot - 1.96 \frac{2310}{6}$

Step 4.3.2.2

Multiply $1.4$ by $- 1.96$ .

$- 2.744 (\frac{2310}{6})$

Step 4.3.2.3

Divide $2310$ by $6$ .

$- 2.744 \cdot 385$

Step 4.3.2.4

Multiply $- 2.744$ by $385$ .

$- 1056.44$

Step 5

Add the results of the summations.

$1509.2 - 1056.44$

Step 6

Subtract $1056.44$ from $1509.2$ .

$452.76$