Author Topic: Calculus Help: Numeric and Geometric explanations? (Read 1537 times)

MegaScientifical · October 10, 2013, 04:22:23 PM

I'm doing work asking me all these questions, and to explain them in words. I remember being told a derivative is the slope of the secant line of a function, but now I'm dealing with tangent lines. I go to look up when a derivative pertains to tangent lines, and I can't even find a place that calls a derivative the slope of a secant. Now I'm just 10x more confused, and it's pissing me off because this is due tomorrow. I have to verbally explain this in excruciating detail, but I don't even understand the basics anymore because everything is contradicting what I know.

Essentially: When is a derivative a secant slope, tangent slope, whatever...?

Edit: http://clas.sa.ucsb.edu/staff/lee/secant,%20tangent,%20and%20derivatives.htm

I don't understand when a derivative is a tangent or a secant. I thought it was the secant slope, but then when is it a tangent? Is it even the secant?

Edit2: Red text at the bottom: http://abcalculus.wikispaces.com/When+does+a+derivative+NOT+exist%3F

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Kalphiter� · October 10, 2013, 05:05:54 PM

Quote from: MegaScientifical on October 10, 2013, 04:22:23 PM

Essentially: When is a derivative a secant slope, tangent slope, whatever...?

The derivative is a tangent line to some point on a curve. ~~The problem may be that this image is a horrible example.~~

The above image is showing, that if you reduce the distance between two points on a secant line, the secant line will approach the same slope as the tangent line. The slope of that tangent line is the derivative.

Treynolds416 · October 10, 2013, 05:27:01 PM

A secant line intersects a curve at exactly two places, a tangent line intersects a curve at exactly one place

Consider the function y = x^2
At x = 1, the slope of the tangent line is 1
For the secant line to have the same slope as the tangent line, it would need to intersect the curve at x = 0 and x = 1

A derivative is always giving the slope of a tangent line, since a derivative is a singular point. A secant line is more of an average between two points. When the distance between the two intersections of a secant line become infinitely close together, it becomes a tangent line.

MegaScientifical · October 10, 2013, 05:27:56 PM

Hmm. Found stuff in my notes, too. Secant is just the equation, but tangent is the limit to the equation.

Secant	=	f[x+h]-f[x] h
Tangent	=	lim f[x+h]-f[x] h->0 h

I did understand Secant = 2 points, Tangent = 1 point. I just wasn't sure when a derivative is one or the other based on given data.

Treynolds416 · October 10, 2013, 05:29:11 PM

A derivative is always evaluated at a single point (that's the purpose of a derivative), so you're always talking about a tangent line

devildogelite · October 10, 2013, 05:31:09 PM

Your mind is gonna be blown when you learn the shortcut for finding a derivative. I know this isn't gonna help you but it'll let you check your derivatives.

Also if you're thinking in terms of physics, a derivative is the velocity.

Kalphiter� · October 10, 2013, 05:35:54 PM

Devil: he has to get through limits first; don't show him that.

Treynolds416 · October 10, 2013, 05:37:09 PM

The power rule didn't really blow my mind. We only spent a week using the formal definition of derivatives before switching to the power rule, and I already knew it from physics anyway

King Leo · October 10, 2013, 05:38:48 PM

Quote from: devildogelite on October 10, 2013, 05:31:09 PM

Your mind is gonna be blown when you learn the shortcut for finding a derivative. I know this isn't gonna help you but it'll let you check your derivatives.

Also if you're thinking in terms of physics, a derivative is the velocity.

How is that supposed to help? He was asking for the definition of a derivative, not how to find the derivative of a power function. Also, it is more correct to say that the derivative of something is the rate of change, and that doesn't just apply for velocity. What is the derivative (rate of change) of distance? Velocity. What is the derivative of velocity? Acceleration. What is the derivative of the mass of water inside a tank? The flow into the tank minus the flow out of the tank.

Choi · October 10, 2013, 05:48:24 PM

Derivatives are pretty much graphs showing the values of the slope of their base graphs at certain x values

Nickelob Ultra · October 10, 2013, 05:56:01 PM

derivatives are the reason I doodle naked Girls in my notebooks during class

Treynolds416 · October 10, 2013, 05:56:40 PM

Quote from: Choi on October 10, 2013, 05:48:24 PM

Derivatives are pretty much graphs showing the values of the slope of their base graphs at certain x values

I'm not sure you completely understand the concept... Derivatives are not necessarily graphs

Quote from: King Leo on October 10, 2013, 05:38:48 PM

How is that supposed to help? He was asking for the definition of a derivative, not how to find the derivative of a power function. Also, it is more correct to say that the derivative of something is the rate of change, and that doesn't just apply for velocity. What is the derivative (rate of change) of distance? Velocity. What is the derivative of velocity? Acceleration. What is the derivative of the mass of water inside a tank? The flow into the tank minus the flow out of the tank.

The power rule is useful for easily evaluating the most basic type of polynomial function, such as how the derivative of x^2 is 2x. As for the rate of change thing, you're right and there's nothing wrong with describing derivatives as such, but I find that it's a limiting definition. For example, you can find the integral of distance, making the derivative of that function the distance. But what would you call that function? The distance function is the rate of change of the unknown function, but it gets more and more confusing the further away you get from known quantities. It's easier imo to just relate it to slope and not think past it

Choi · October 10, 2013, 05:59:13 PM

Quote from: Treynolds416 on October 10, 2013, 05:56:40 PM

I'm not sure you completely understand the concept... Derivatives are not necessarily graphs

I know they're not only graphs

I didn't want to tell a confusing definition

King Leo · October 10, 2013, 06:05:27 PM

Quote from: Treynolds416 on October 10, 2013, 05:56:40 PM

I'm not sure you completely understand the concept... Derivatives are not necessarily graphs
The power rule is useful for easily evaluating the most basic type of polynomial function, such as how the derivative of x^2 is 2x. As for the rate of change thing, you're right and there's nothing wrong with describing derivatives as such, but I find that it's a limiting definition. For example, you can find the integral of distance, making the derivative of that function the distance. But what would you call that function? The distance function is the rate of change of the unknown function, but it gets more and more confusing the further away you get from known quantities. It's easier imo to just relate it to slope and not think past it

I know, I know. But it is often (at least for me) helpful to visualize abstract concepts such as this, instead of just thinking about it purely mathematically. It makes it easier to remember and understand.
That's why I don't like stuff that can't be understood by visualisation, such as quantum mechanics (bleh!) and (to some extent) statistics...

devildogelite · October 10, 2013, 06:06:50 PM

Quote from: King Leo on October 10, 2013, 05:38:48 PM

How is that supposed to help? He was asking for the definition of a derivative, not how to find the derivative of a power function. Also, it is more correct to say that the derivative of something is the rate of change, and that doesn't just apply for velocity. What is the derivative (rate of change) of distance? Velocity. What is the derivative of velocity? Acceleration. What is the derivative of the mass of water inside a tank? The flow into the tank minus the flow out of the tank.

forget man, I said it'll help you check most of what your doing. When I was first learning calc knowing that shortcut helped me check my answers quite a bit. It seems like he isn't comfortable with derivatives so I figured that would help.

I'm sorry I didn't say it was rate of change. I always just think velocity because I learned derivatives in Physics. I've been doing derivatives for quite some time now so I know how they work. I though maybe if he thought velocity it would give him a better understanding.

Sure looks like I screwed the pooch on trying to explain derivatives.