Question

Commuting with constant multiples: What does it mean to say that derivatives commute with constant multiples? Do that limits commute with constant multiples? Does that sum write in sigma notation commute with constant multiples? Express your answers in words and algebraically.

Short answer

Ans: The summation of the constant multiplied by the function is the same as that of the constant multiplied by the summation of that function.

$\sum kf\left(x\right)-k\sum f\left(x\right)$

Step-by-step solution

Step 1. Given information.

given,

Consider the derivatives and limits.

Step 2. Derivatives commute with constant multiples:

The derivative of the constant multiplied by the function is the same as that of the constant multiplied by the derivative of that function.

$\frac{d}{dx}\left(kf\right(x\left)\right)=k\frac{d}{dx}f\left(x\right)$

Step 3. Limits commute with constant multiples:

The limit of the constant multiplied by the function is the same as that of the constant multiplied by the limit of that function.

$\underset{x\to a}{lim} \left(kf\right(x\left)\right)=k\underset{x\to a}{lim} f\left(x\right)$

Step 4. Sums are written in sigma notation commute with constant multiples.

The summation of the constant multiplied by the function is the same as that of the constant multiplied by the summation of that function.

$\sum kf\left(x\right)-k\sum f\left(x\right)$