What is the difference between an 'increasing gradient' and a positive gradient?

In summary, the terms "positive gradient" and "increasing gradient" do not mean the same thing. A positive gradient means that the slope of the graph is positive, while an increasing gradient means that the slope is increasing over time. To determine if a gradient is increasing or not, one must look at the second derivative of the function.
  • #1
Dramacon
14
0

Homework Statement


f(x)= 3+6x-2x^3

(a) Determine the values of x for which the graph of f has positive gradient
(b) Find the values of x for which the graph of f has increasing gradient

Homework Equations


I had originally thought the two terms meant the same thing, but when I checked the answers at the back of the book, they gave two different answers.


The Attempt at a Solution


Isn't a positive gradient an increasing one?
 
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  • #2
An increasing gradient means that the gradient itself is increasing.
 
  • #3
So when a line is at more than 45 degrees, you mean?
 
  • #4
Or do you mean increasing from one stage to the next?
 
  • #5
If the gradient of the gradient is positive, then it is an increasing gradient.
Do you know how to check for that?
 
  • #6
Dramacon said:

Homework Statement


f(x)= 3+6x-2x^3

(a) Determine the values of x for which the graph of f has positive gradient
(b) Find the values of x for which the graph of f has increasing gradient

Homework Equations


I had originally thought the two terms meant the same thing, but when I checked the answers at the back of the book, they gave two different answers.


The Attempt at a Solution


Isn't a positive gradient an increasing one?

suppose our function was g(x) = 2x + 3.

at any given point, the gradient (slope of the graph) is constant, it is 2.

note that g'(x) = 2 is positive, but it ISN'T increasing, it's flat.

to see whether or not the gradient is increasing/decreasing/neither, you need to find the gradient of the gradient.

in terms of derivatives, this means you need to look at the second derivative, to tell whether the first derivative is increasing, decreasing, or "flat". note that these are "local" properties, the answers you get depend on "which x" you look at.
 
  • #7
Ah, I see! :) Thank you! This makes so much more sense now.
 

1. What is the meaning of an "increasing gradient" in science?

An increasing gradient refers to a slope or incline that is becoming steeper or higher in value over time. In science, this often refers to the rate of change of a variable or the direction of a force.

2. How is an increasing gradient different from a positive gradient?

While both an increasing gradient and a positive gradient indicate a growth or increase in value, a positive gradient specifically means that the value is increasing in a positive direction, such as moving upwards on a graph or in a positive direction along an axis.

3. Can a gradient be both increasing and negative?

Yes, a gradient can be both increasing and negative. This would mean that the value is decreasing in a negative direction, such as moving downwards on a graph or in a negative direction along an axis.

4. How do scientists measure an increasing gradient?

Scientists can measure an increasing gradient by determining the change in value over a specific interval, such as time or distance. This change in value is then divided by the corresponding change in time or distance to calculate the gradient.

5. What are some real-life examples of an increasing gradient?

Some real-life examples of an increasing gradient include the rate of temperature increase in a heating system, the velocity of a car accelerating, and the growth of a population over time. These examples all exhibit a trend of becoming steeper or higher in value over time.

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