Question 1

I am sorry I dont get this straight ahead, but why do we shift it to the right when we have -c and not to the left? Sal doesnt get very deep on the explanation of this...

Accepted Answer

I'll try to explain but I don't claim to be a master at math nor teaching.

so, the question is, since it is MINUS c why does it shift to the RIGHT. Shouldn't it shift to the left since the x-axis is a number line and negative numbers go towards the left?

That is an extremely easy mistake to make because it does kinda make sense. However, try to think of it like this: When you plug an x-value into a function, the x value never changes only  the y-value does. In other words, the input to a function is always going to be whatever you input, no matter what.

For example, take y=x^2 this is a function. You put in 2 and you get out 4. The 2 you put in(x-value) is always 2. The y-value is the one that got changed. The x-value was squared to change it into a y-value.

Here is where this will make sense(hopefully). lets take that function and shift it. Now lets look at y=(x-2)^2 we think it should shift to the left because of x-2. We think this because we mistakenly believe that when we put an x value into that function then that x value will move 2 to the left. As stated before, only y values move. So, plugging any number(aka x) into this will give us the y value that would appear at x minus 2. Looking at our function, y=(x-2)^2 if we plug in 0 it will spit out 4. 4 is the y value we would have gotten at x=-2 if this function was not shifted.  
So essentially, the -2 shifts it to the right because it takes x, then subtracts 2 from it, then goes through the rest of the function(in this case that means squaring it), then puts that y value at the original x value, which is 2. Since it does this for every x value that means the whole entire function is shifted to the right.  
To condense it more and talk about Sal's function, f(x-c), pretend that when you take any x, the negative c shifts it to the left by c then takes whatever y value is at x-c and puts that y value at x.

Question 2

Why is there a dx at the end of each function?

Accepted Answer

Did you finish the section just prior to this on Riemann Sums?

Recall we were calculating the area under the curve via subdividing the region into rectangles whose height was f(x) and whose width was Δx. We found that the accuracy of the estimate of the area got better and better as the number of subdivisions increases, in which case the width, Δx, is obviously decreasing.

It made intuitive sense, then, that to get better accuracy, we need to keep taking narrower and narrower slices, that is, taking the limit of sum of the rectangles as the width, Δx, approaches zero. Once we say that, then we are dealing with Δx which is infinitesimal (cannot be measured). We denote the infinitesimal Δx as dx.

When we were summing via Riemann approximations we used the symbol ∑ to indicate the summing of each of the slices. The integral symbol ∫ is actually an elongated S which stands for sum and is used when summing infinitesimals. So the only difference between the Riemann sum form ∑f(x_i)Δx_i and the integral form ∫f(x)dx, is that the latter is summing the areas in infinitesimally thin slices, where as the former is made up of slices whose widths can be measured, either way, the slice area has a height component, f(x) and a width component, dx for infinitesimals, or Δx if not.

Question 3

is f(x-2) same as y=x-2?

Accepted Answer

No. f(x-2) means, when you are given an x, first subtract 2, and then put that value into your function. Say your function is f(x) = x^2, then, f(x-2) = (x-2)^2; if your function were f(x) = x + 5, then f(x-2) = x - 2 + 5 = x + 3.

Question 4

Can someone please explain to me why the function is shifted to the right if it is a negative sign? I know this as a general rule but I am stuck on why this happens

Accepted Answer

Let's say we have a function such as y = x²
So, one of the most notable features of this parabola is its vertex, in this case its lowest point, which happens to be at (0, 0).

Let's compare this function to y = (x- 2)² which is also a parabola.  Where is its vertex?  The lowest possible point of the parabola will occur when x - 2 equals zero.  That happens when x is 2 with this equation.  All other values will be greater than zero, as we can see below.
From the equation, which is in the "vertex form" y = (x- `2`)² ` `  we can directly read that the vertex is at (2, 0) or we can plot the curve by making a value table:
x  |   y =  (x- 2)² 
`-1      9`
`0      4`
`1       1`
`2      0`   Lowest value of the function:  the vertex
`3      1`
`4      4`

The values show that the lowest point of this parabola is at x = 2, with coordinates of (2, 0)
So, even though the equation shows a negative, the feature we are watching has moved to the right.

Question 5

If the f(x) shifts right after you subtract C, then why do points A and B shift right when you ADD C?

Accepted Answer

By doing (x-2) you're changing the input x, not f(x).  Basically by changing the input x that goes into the equation negatively, you're shifting it all to the right.  A good example that helps me is to imagine y=x^2 as f(x).  If you then do f(x-2) you get the equation y=(x-2)^2 instead of y=x^2-2.

Course: Integral Calculus (2017 edition) > Unit 1

Definite integral of shifted function

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