# What is meant by the slope of a function?

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I think what you are asking about is the idea of a function having a slope at a point, which is often defined as the limit of the slope of a secant through two points as one point approaches the other.

graph{(y-x^2)((y-1) - 2(x-1)) = 0 [-1.95, 4.21, -0.576, 2.504]}

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The slope of a function refers to the rate at which the function's output (dependent variable) changes with respect to its input (independent variable). Geometrically, it represents the steepness of the function's graph at a given point.

Mathematically, the slope of a function ( f(x) ) at a specific point ( x = a ) is defined as the derivative of the function at that point, denoted as ( f'(a) ) or ( \frac{dy}{dx}|_{x=a} ).

The slope can also be interpreted as the ratio of the change in the function's output to the change in its input within a given interval. This is expressed by the formula:

[ \text{Slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x} ]

Where ( \Delta y ) represents the change in the function's output and ( \Delta x ) represents the change in its input.

In summary, the slope of a function provides information about how the function is changing as its input varies, and it is fundamental in calculus for understanding rates of change and the behavior of functions.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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