Question

Evaluate the function at the given value of the independent variable. simplify the results. f(x) = x3 f(x + δx) − f(x) δx (δx ≠ 0)

Answer 1

To evaluate the function f(x) = x^3 for the expression (f(x + δx) - f(x))/δx, we find f(x+δx) and subtract f(x), then divide by δx to get the average rate of change of the function, simplifying the result to 3x^2 + 3xδx + (δx)^2.

Step-by-step explanation:

The student's question asks to evaluate the function f(x) = x^3 at the value of f(x + δx) - f(x) over δx, given that δx is not zero. This is essentially looking for the average rate of change of the function over the interval from x to x+ δx. The first step is to substitute x+δx into the function to obtain f(x+δx). Then, the next step involves subtracting f(x) from that result and dividing by δx to simplify the expression, as requested. However, it is important to note that because δx is not zero, we are not finding the derivative of f(x), just an average rate of change.

To provide a detailed solution, we evaluate f(x+δx) by cubing the expression x+δx to get (x+δx)^3. Next, we simplify it to x^3+3x^2δx+3x(δx)^2+(δx)^3. Then subtracting f(x) = x^3 yields 3x^2δx+3x(δx)^2+(δx)^3. Finally, by dividing the entire expression by δx, we simplify the result to 3x^2 + 3xδx + (δx)^2. This expression represents the average rate of change of the function f(x)=x^3 over the interval [x, x+δx].

Answer 2

we are given

`f(x)=x^3`

Firstly, we will find f(x+delta x)

`f(x+\delta x)=(x+\delta x)^3`

now, we can plug this in formula

`(f(x+\delta x)-f(x))/(\delta x)`

`(f(x+\delta x)-f(x))/(\delta x)=((x+\delta x)^3-(x)^3)/(\delta x)`

now, we can simplify it

`=(x^3+(\delta x)^3+3x^2\delta x+3(\delta x)^2x-x^3)/(\delta x)`

now, we can simplify it

`=((\delta x)^3+3x^2\delta x+3(\delta x)^2x)/(\delta x)`

`=(\delta x ((\delta x)^2+3x^2+3(\delta x)x))/(\delta x)`

we can cancel it

`=(\delta x)^2+3x^2+3(\delta x)x`...........Answer