Graphs and Differentiation

Graphs and Differentiation

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The Significance of Gradients in Graphs

A graph's gradient indicates its steepness at a specific point, making it a crucial aspect of comprehending functions. To better understand gradients, let's consider a simple linear equation:

For instance, let the gradient be 3 for the graph of y=3x. But what happens when the graph is not a straight line? In such cases, we have a function of the gradient rather than a single point with a real value. Let's take an example to illustrate this:

If y=x^2, what is the gradient at the point (1,1)?

A positive gradient value signifies an upward slope, whereas a negative value represents a downward slope. The blue line indicates a negative gradient, the green line a zero gradient, and the purple line a positive gradient. However, there are instances where the gradient's value is neither positive nor negative but zero. This occurs when the graph has a maximum or minimum point.

For example, in y=x^2, the minimum and maximum points are both at (0,0). This can be verified by calculating the second derivative. If the value is positive, it's a maximum point, if negative, it's a minimum point, and if it's neither, it's a point of inflection.

To gain a better understanding, let's look at the graph of y=x^3, which has a point of inflection at (0,0).

So how do we determine the nature of a stationary point? Let's take an example to demonstrate this.

What is the minimum point of y=3x^2?

To determine the stationary point's nature, we need to calculate the second derivative. Substituting the values from the quadratic formula, we get 2. Since the second derivative is positive, this is a minimum point. Geometrically, this means that the gradient at this point is 0.

The key takeaway is that the gradient represents the slope of the curve at a particular point, and it is the equation of the tangent line at that point. A tangent is a line that touches the curve at one point without intersecting it, as shown in the image below.


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