Evaluate the Summation sum from x=0 to 1 of 2x-4

Solution:

Step 1: Write out the terms of the summation

Calculate the terms of the series by substituting the values of $x$. For $x = 0$ and $x = 1$, the terms are $2 \cdot 0 - 4$ and $2 \cdot 1 - 4$ respectively.

Step 2: Perform the calculations

Step 2.1: Apply multiplication

For the first term, multiply $2$ by $0$ to get $0$, and for the second term, multiply $2$ by $1$ to get $2$.

The series now looks like: $0 - 4 + 2 - 4$.

Step 2.2: Combine like terms

Combine the constants by adding and subtracting them in sequence.

First, subtract $4$ from $0$ to get $-4$.

Next, add $2$ to get $-2$.

Finally, subtract $4$ to get the sum: $-6$.

The final result of the summation is $-6$.

Knowledge Notes:

To evaluate the summation of a series, you follow these steps:

1. Expansion: Write out each term of the series by substituting the values of the variable into the given expression.

2. Simplification: Perform the operations indicated in each term, such as multiplication, addition, or subtraction.

3. Combination: Add or subtract the simplified terms to find the total sum.

In this problem, the series is a simple arithmetic expression involving multiplication and subtraction. The summation is over a finite and small range, so it can be done by direct calculation without the need for more complex summation formulas or techniques.

The expression $2x - 4$ is a linear function of $x$. When evaluating the summation of a linear function over a range, you can substitute each value of $x$ within the range and then perform the arithmetic operations to find the sum.

In LaTeX, multiplication is often denoted by $\cdot$, subtraction by $-$, and addition by $+$. When writing series or summations, the sigma notation $\sum$ is used, but in this case, since the range is small, it's straightforward to write out each term explicitly.