Curve Sketching of f = x^2, f' = 2x

[Miike012]

f= x^2
f' = 2x


f'> 0 in (0, ∞)... f = y increases on this interval
f' < 0 in (-∞,0) f = y decreases on this interval


I don't understand... it is saying that f decrease on (-∞,0) but obviously its not because (∞)^2 = ∞ which is increasing...

Does this mean that the function is decreasing as as x approaches zero from the left?? If it did that would make more sense to me..

 

[Ray Vickson]

f= x^2
f' = 2x


f'> 0 in (0, ∞)... f = y increases on this interval
f' < 0 in (-∞,0) f = y decreases on this interval


I don't understand... it is saying that f decrease on (-∞,0) but obviously its not because (∞)^2 = ∞ which is increasing...

Does this mean that the function is decreasing as as x approaches zero from the left?? If it did that would make more sense to me..

It means that if x1 < x2 < 0 then f(x1) > f(x2). For example, what is f(-10)? What is f(-5)? Is -10 less than -5?

RGV

 

[Miike012]

f(-10) = 100 What is f(-5) = 25... so how does that mean f is decreasing on (- inf , 0)..
But as x gets larger negative y increases... so obviously y is not decreasing on (- inf , 0).

 

[Ray Vickson]

f(-10) = 100 What is f(-5) = 25... so how does that mean f is decreasing on (- inf , 0)..
But as x gets larger negative y increases... so obviously y is not decreasing on (- inf , 0).

As x *increases* from -10 to -5, f decreases from 100 to 25. That is what we mean by a decreasing function. As I have said already, f decreases if x1 < x2 gives f(x1 ) > f(x2), and that is exactly what happens for negative x values. What part of that statement do you not understand?

RGV

 

[Miike012]

I thought it meant that the y-values constantly decrease..

For instance if [x_n , x₀] where x₀ <= 0

Then as x ---> x_n that f(x) --> smaller and smaller y values...
Look at my picture...

 

[ehild]

For negative quantities, greater magnitude means less value.
Think: when you spend money - a sum of m - it is like a negative income. When are you richer: if your income is -100 or if it is -5?

ehild

 

[Ray Vickson]

I thought it meant that the y-values constantly decrease..

For instance if [x_n , x₀] where x₀ <= 0

Then as x ---> x_n that f(x) --> smaller and smaller y values...
Look at my picture...

We increase something by adding a positive quantity to it. If it starts negative, adding a positive quantity to it either makes the result "less negative" (i.e., smaller in SIZE) or else positive (which may be either smaller in size or larger in size, depending on details). Similarly, we decrease something by adding a negative amount to it, or equivalently, by subtracting a positive amount from it. If it starts negative, decreasing it makes the result "more negative" (larger in SIZE). "Increasing" means we move to the right on the real number line, while decreasing means moving to the left. It does not matter where we start from. You need to stop confusing magnitude and position. Words in mathematics have specific meanings, and these may not always agree 100% with everyday usage.

RGV