Calculus and Slope

[Jason76]

The slope for any linear function is the same no matter which x, y points are used. However, with non-linear functions the slope cannot be the same for all points. In fact, it's always different. Threfore, we find the generalized non-linear function slope (derivative) that allows us to find slopes for individual points. In order to find each individualized slope, we simply plug tje specific x value into the derivative (no need even to use the y value). And of course, you could use "change in x" notation, but using things like the power rule does the same thing.

From the "change in x" notation (not power rule type differentation) we realize the generalized non-linear slope (derivative) is a limit. This is because when making the tangent line we use one of the points on the function and then choose another point which is close, but doesn't equal 0. Anyhow, by picking the random point and manipulating the tangent line, then we produce a generalized derivative, which (as we said before) allows us to find the slope for any individual point.

Is this on the right track?

 

[JeffM]

The slope for any linear function is the same no matter which x, y points are used. However, with non-linear functions the slope cannot be the same for all points. In fact, it's always different. Threfore, we find the generalized non-linear function slope (derivative) that allows us to find slopes for individual points. In order to find each individualized slope, we simply plug tje specific x value into the derivative (no need even to use the y value). And of course, you could use "change in x" notation, but using things like the power rule does the same thing.

From the "change in x" notation (not power rule type differentation) we realize the generalized non-linear slope (derivative) is a limit. This is because when making the tangent line we use one of the points on the function and then choose another point which is close, but doesn't equal 0. Anyhow, by picking the random point and manipulating the tangent line, then we produce a generalized derivative, which (as we said before) allows us to find the slope for any individual point.

Is this on the right track?

It is on the right track, but I would word it differently. For one thing, you are really talking about differentiable functions, and linear functions are differentiable. You are making an unnecessary distinction between linear and non-linear differentiable functions.

For any differentiable function F(x), linear or not, the derivative F'(x) is a new function that specifies the limit of the ratio of the change in F(x) to the change in x, which is also the slope of the line tangent to the graph of F(x). In the case of a linear function, that limit and that slope do not change at different values of x, but in general that limit and that slope do change at different values of x. The derivative provides a formula for finding that limit and slope. The formula is very simple for a linear function, but it is still a formula.

Furthermore, I have no idea what you are saying about the power rule. To save time, theorems have been developed to find derivatives of many classes of functions. The power rule is one of those theorems. The fact that you use the power rule to find a derivative does not change what the derivative represents. Maybe all you are saying is that it is not obvious when using generalized rules like the power rule that they arise from a limiting process.

See what others have to say.

 

[Jason76]

You can differentiate a linear function, but what's the use? The point of calculus was to solve problems regarding non-linear functions. For instance, one such problem is "instantaneous rate of change".

Yes, as you were saying, all derivatives are limits, but when you don't see the limit in time saving procedures like the power rule.

In the case of a linear function, that limit and that slope do not change at different values of x, but in general that limit and that slope do change at different values of x.

Your saying that, in the case of a linear function, the slope and limit doesn't change. I agree there. But then you say they do change. What do you mean?

 

[JeffM]

You can differentiate a linear function, but what's the use? The use is to get you to focus on the fundamental distinction. Calculus applies only to differentiable functions; distinguishing between linear and non-linear functions does not tell you where calculus applies. The point of calculus was to solve problems regarding non-linear functions. It is true that you seldom if ever need to use calculus when dealing with linear functions; basic algebra will do. But this misses the point that math is a generalizing process, and calculus is generalizing concepts first learned with respect to linear functions, which form one tiny province in the empire of differentiable functions. For instance, one such problem is "instantaneous rate of change".

Yes, as you were saying, all derivatives are limits, but when you don't see the limit in time saving procedures like the power rule. So what? I still am not sure what you are driving at.



Your saying that, in the case of a linear function, the slope and limit doesn't change. I agree there. But then you say they do change. What do you mean? I am drawing a distinction between differentiable functions, which usually have derivatives with different values at different values of the argument, and the special case of linear functions.

It's late. If you find my opinion unhelpful, simply ignore it.

 

[Jason76]

It's late. If you find my opinion unhelpful, simply ignore it.

Possibly this thread is thinking about "slope and calculus" too deeply. The aim is being able to solve problems, and there is enough information to do that.

 

[daon2]

You said for a nonlinear function "In fact [the slope] is always different" which is not true. For example, f(x)=x^3 has a derivative which is an even function so the slope of f(x) is the same at x=a and x=-a for all real numbers a. You can say that if the concavity of a function never changes, then it must be true.

Since we're talking about the usefulness of derivatives I'll add some possibly irrelevant information: The point of a derivative is to study how a function behaves locally (close to some x-value: a). The derivative at a point (a,f(a)) gives you a tangent line, that is, a linear approximation to f(x) when x is close to a. But there are other (better) approximations. Using "secant parabolas" for example will give rise to tangent parabolas, and also for cubics and so on. When you jump into "calc 2" land you will probably study taylor polynomials which are exactly "tangent polynomials" to your function. The higher the degree of this polynomial, the more you know about how your function is changing around your x-value.

 

[Jason76]

You said for a nonlinear function "In fact [the slope] is always different" which is not true. For example, f(x)=x^3 has a derivative which is an even function so the slope of f(x) is the same at x=a and x=-a for all real numbers a. You can say that if the concavity of a function never changes, then it must be true.

Since we're talking about the usefulness of derivatives I'll add some possibly irrelevant information: The point of a derivative is to study how a function behaves locally (close to some x-value: a). The derivative at a point (a,f(a)) gives you a tangent line, that is, a linear approximation to f(x) when x is close to a. But there are other (better) approximations. Using "secant parabolas" for example will give rise to tangent parabolas, and also for cubics and so on. When you jump into "calc 2" land you will probably study taylor polynomials which are exactly "tangent polynomials" to your function. The higher the degree of this polynomial, the more you know about how your function is changing around your x-value.

Right, I agree there. So the point of derivatives is "finding the slope of curved functions or parts of a function which happen to be curvy". Saying that derivatives were specifically made for nonlinear functions is misleading.

 

[daon2]

Right, I agree there. So the point of derivatives is "finding the slope of curved functions or parts of a function which happen to be curvy". Saying that derivatives were specifically made for nonlinear functions is misleading.

Who said that? It makes sense to include linear functions as things to study, sure. If nothing else, they are a good stepping stone into the topic of "rates of change." Most functions are not differentiable. Of those that are, most are not polynomials. Farther still, most polynomials are not linear (at least, intuitively). I would say the aim is to discuss functions in the most generality appropriate for the class setting and one would hope that linear functions become a triviality after some time. So they have their place, but should not be central objects of study.