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[frostshotgg]

Yes. f'(2) = 4. You took the most impossibly circuitous route to get there though. Keeping derivative = slope of tangent in mind, and knowing that the slope of any equation in the form of y = mx + b is m, you know that f'(x) at that point is m. No need to do any math. It should take you a second or two max to identify that.

[Spehss _]

Yes. f'(2) = 4. You took the most impossibly circuitous route to get there though. Keeping derivative = slope of tangent in mind, and knowing that the slope of any equation in the form of y = mx + b is m, you know that f'(x) at that point is m. No need to do any math. It should take you a second or two max to identify that.

To determine what f'(2) is by y=mx+b wouldn't I have to calculate what the equation of f'(x) is?

It sounds like you're saying f'(x) (the derivative, right?) is equal to the tangent equation given, y=4x-5, so f'(x)=4x-5 and f'(2)=4(2)-5 which is 3 which was the same as f(2).
No wait. Derivative=slope of tangent, so f'(x)=m and m=4 so f'(x)=4 so f'(2)=4 or f'(3)=4 or f'(ANYTHING)=4?

I thought f'(x)(ie derivative) was a separate equation from f(x), not just a single number.

[frostshotgg]

The slope of the tangent line changes throughout a curve. The tangent line my be y = 4x-5 at f(2), but then y = 16x+7 at f(3). So then f'(2)=4, and f'(3)=16. But by knowing the tangent line at a point means you know the derivative at that point, just not necessarily at all of the points.

[Helgoland]

Stop thinking in equations. Think in functions instead: A function f is just a machine that associates to every number x some other number f(x). The function's derivative then is just another machine, associating to every number x some number f'(x). We can use the same idea even more generally and assign other objects to other objects too - there's an example in the next paragraph.

This may seem like a trivial difference, but it's really not: Equations are essentially just statements that some relation holds. Functions are independent mathematical objects, which we can work with without concerning ourselves with numbers. As an example, you could consider the process of derivation a functionassociating to every (differentiable) function f its derivative f'.

To be more concrete: Your f' is a function just like f is. The equation y=4x-5 (actually you should write that as f'(x) = 4x - 5, but since you're doing a couple sort-of graphical arguments I guess y is fine as well) tells you what relationship there is between a number x and the number f'(x) assigned to it by the function f. The function itself is a distinct entity from the equation! You can see this by checking that y = 2^2 * x - 5, which is technically speaking a different equation, gives the same relationship.
f'(x) = m holds as long as you use the m calculated for that specific x. Since m depends on x - the slope of the tangent is not constant, after all - m is really a function, and it would be better to write f'(x) = m(x). But this basically just means that the two functions are the same, which is a more fancy way of saying derivative = slope of tangent.




If all that is too abstract for you, here's your concrete error: m is not fixed, but varies as x varies. f'(x) = m only holds as long as you're talking about the m calculated for that specific point.

Pseudo-edit: Damn ninjas getting in the way of my ranting... By the way, do tell me about pedagogical errors I'm making - I'll be tutoring some physicists next semester, and I want to improve my relevant skills.

[Spehss _]

The slope of the tangent line changes throughout a curve. The tangent line my be y = 4x-5 at f(2), but then y = 16x+7 at f(3). So then f'(2)=4, and f'(3)=16. But by knowing the tangent line at a point means you know the derivative at that point, just not necessarily at all of the points.

That's what I thought. I probably misinterpreted your first post a bit.

Side question, am I supposed to be getting "undefined" occasionally for slopes? In two problems now I wind up dividing by zero when calculating the slope for tangent lines and I can't tell if that's supposed to happen or if it means I messed up somewhere.

[Bauglir]

Consult the original equation. If it has a discontinuity at the point you're trying to find a tangent line for (for example, x=0 in 1/x or x²/x), you should expect something undefined. You should also expect an undefined derivative at "sharp" corners, like x=0 for |x| (absolute value of x, if you haven't seen the notation).

[Spehss _]

Consult the original equation. If it has a discontinuity at the point you're trying to find a tangent line for (for example, x=0 in 1/x or x²/x), you should expect something undefined. You should also expect an undefined derivative at "sharp" corners, like x=0 for |x| (absolute value of x, if you haven't seen the notation).

I was about to post saying that both problems only provided the functions/equations, not graphs to look at and check, then I remembered graphing calculators are a thing. Duh. Upon checking the graphs I think I definitely messed up somewhere and I'll have to redo those.

Thanks for the help, guys, I've been having a somewhat harder time with calculus homework than I have in past math classes. On an unimportant and unrelated note, my confidence in my intelligence has taken a nosedive during my brief time in this thread. :p

[Bauglir]

If it's any consolation, I've noticed that a lot of math classes, especially Calculus, present the material in the most obtuse possible way. I think part of it is that many teachers/textbooks will try to use the same strategies that get people through primary school to teach this material. Calculus is about when it's much more important to grok the concepts the numbers and symbols represent, and the usual rote memorization of rules, formulas, and special cases is not a useful foundation. They work better as shortcuts, which is why a good Calculus class teaches you the definition of a derivative (for example) instead of the convenient shenanigans everybody really uses, instead of the other way around.

I cannot recommend enough the Calculus Revisited resources located on this page when you want to review something. The lectures spend most of their time walking through conceptual explanations and proofs instead of listing facts.

EDIT: By the way, even without a graph, you can quickly check on discontinuities and that sort of thing by finding out if there are any values of variables that can break the equation. Look for values that can cause a division by 0 as your most obvious bet, so that if there's a denominator of (x-1) floating around your equation, x=1 will be what you need to watch out for. Also, Helgo's explanation of the concept of a function up above is a good one, and important to grokking calculus.

[i2amroy]

If it's any consolation, I've noticed that a lot of math classes, especially Calculus, present the material in the most obtuse possible way. I think part of it is that many teachers/textbooks will try to use the same strategies that get people through primary school to teach this material. Calculus is about when it's much more important to grok the concepts the numbers and symbols represent, and the usual rote memorization of rules, formulas, and special cases is not a useful foundation. They work better as shortcuts, which is why a good Calculus class teaches you the definition of a derivative (for example) instead of the convenient shenanigans everybody really uses, instead of the other way around.

Strange, I actually learned best from the standard "memorize all twenty derivative formulas" type of approach. Being taught the definition of a derivative was nice to know, I guess, but learning wise it was worth absolutely nothing to me other than as another random fact. (I learned integrals by just taking the 20 or so memorized derivative formulas and running them backwards for the most part ).

[Bauglir]

Hm, fair enough. I might be being overly dogmatic. Still, while the rules are what I actually use day-to-day and I can function just fine without the definition, the definition is a major part of why I'm able to understand what "derivative" means, even if most days I couldn't quote you the relevant formula.

[frostshotgg]

As far as calculus goes, being able to take the derivative/integral of something is 10% of the work at most. Usually it's 1 point of the process, max unless a problem involves differentiating/integrating multiple times, with the rest being the logic behind it. Between that fact, and the fact that you get a TI-89 which can integrate/differentiate for you now, it's just not important to do more than memorize derivative rules. No sane person can reasonably expect you to be able to do d/dx(17e^xcos3x) with the limit definition of a derivative.

Also integrals are literally rule memorization. The reason some functions are nonintegrable is because we literally don't have any rules for them, because all we can do to integrate things are either A, Riemann summs, or B, know what derivative it happens to be the inverse of.

[Bouchart]

Most of calculus is just algebra and some trigonometry.

[bahihs]

Most of calculus is just algebra and some trigonometry.

I actually understand calculus best as geometry rather than the symbolic approach of those. It's also prettier.

[Spehss _]

More calculus and derivatives. Problem wants me to calculate f'(x) by graphing f(x) on a calculator and zooming in on the x, then use symmetry to deduce the values of f'(x), then use the first two results to guess an equation for f'(x) and justify our answer.

If f(x)=x^(2) then wouldn't f'(x)=x? The function gives the square of x which means the slope is x so that x is multiplied by x. Graphing seems unnecessary.

[frostshotgg]

Since f'(x^2) = 2x, not 1x, it would seem that graphing is in fact necessary for you to grasp whatever it wants you to.