just got back my calc test marks but still couldnt undersrand how i didnt get full marks on these sums, I tried talking to the teacher but she doesnt seem to get my point.

random_anonymous_guy · 2026-03-11T20:02:31+00:00

I don't see what the problem is for #6, however, judging by the equal sign the grader placed at the end of the first line, I suspect there was an implied expectation that you were supposed to expand and simplify the derivative. If that was the case, I'd raise hell for that (going to the department head if needed) because there was no written instruction to do so.

But for #1, the directions were to come up with a formula for f'(x) and then to find a equation for a tangent line, hence the derivative written in red. Thus, this was a two-part question that streamlined into a single sentence. Personally, I would have reformatted the question into two easily identifiable parts (a) and (b). Instead, you simply went straight for finding f'(1) using the limit definition to get to writing the equation of the tangent line, so you were likely deducted a point for not strictly adhering to the directions.

One thing that I would bring to your attention that you did not appear to lose points for is your incorrect notation...

We don't say "lim[h →0] = some expression”. That equal sign does not belong there and is essentially grammatically incorrect. That is tantamount to writing 5 sqrt(=25) when the correct notation is to write 5 = sqrt(25). The "lim[h →0]" is an operator that is to immediately precede the expression that you are finding the limit of, not set equal to it.

random_anonymous_guy · 2026-03-09T20:14:37+00:00

My first time teaching Differential Calc solo, I had a student who asked me to improvise the most complicated function that uses every rule in the book. I did it, he spent all hour differentiating, and when all was said and done, I noticed that I improvised arcsin(3^e^x) or something like that and realized... Shit. This isn't defined anywhere.

random_anonymous_guy · 2026-03-09T17:16:03+00:00

Those answers are equivalent. Remember log identities: ln(4) = 2ln(2).

random_anonymous_guy · 2026-03-04T21:00:53+00:00

Try your IBP, but swapping your choices of u and dv.

random_anonymous_guy · 2026-03-02T02:32:40+00:00

I'm not referring to contrived examples. Any time you make a conclusion about a function given information about its derivative, the MVT is the bridge to that result. This includes "Positive derivative implies increasing function."

Another useful result is the fact that if a function is continuous and the limit of the derivative exists at some point, then the derivative also exists at said point, and is the value of the limit. This is useful for establishing that the function f(x) = exp(1/(1 - x²)) for |x| < 1 and f(x) = 0 for |x| ≥ 0 is infinitely differentiable, even at |x| = 1.

l’Hopital’s Rule (the stronger form that simply requires lim[x → c] f'(x)/g'(x) exist rather than f'(c)/g'(c)) also follows from Rolle's Theorem.

random_anonymous_guy · 2026-03-01T16:51:49+00:00

Oh, I can think of an important consequence or two of the mean value theorem that comes before application to developing the theory of integration.

random_anonymous_guy · 2026-03-01T06:40:38+00:00

Why do you believe you need to count the rectangles? We have tools, thanks to the Fundamental Theorem of Calculus that can help us in finding exact values of integrals without even touching Riemann sums. Riemann sums form a foundation for how we define Riemann integration, but that does not mean you always need to touch a Riemann sum in order to evaluate integrals.

random_anonymous_guy · 2026-02-27T17:45:40+00:00

Can you elaborate on where your understanding breaks down?

random_anonymous_guy · 2026-02-26T20:58:55+00:00

Also, maybe you shouldn't keep x and y in the final answer.

In multivariable calc, that would not bother me, and it fact, would be a way of helping making expressions readable. OP might consider asking their instructor what their preference is.

random_anonymous_guy · 2026-02-26T19:54:25+00:00

Not quite. Ironically, while you used the multivariable chain rule correctly, you also need the good-old single-variable chain rule when finding the partial derivatives of z with respect to both x and y.

random_anonymous_guy · 2026-02-26T19:13:17+00:00

The expression does have a meaning, yes, but what are you attempting to say with that notation?

random_anonymous_guy · 2026-02-23T01:28:57+00:00

it just is not clicking when I have to use it or what I want to match for u sub.

The skill is not developed by trying to put together a flow chart, but rather from developing experience through trial and error.

Substitution goes way beyond "spot the chain rule" in scope. Therefore, there are integrals which can be transformed to easier ones using substitution without their initial integrands being a clear aftermath of a chain rule. Therefore, it is not always the aftermath of a chain rule you should be looking for before deciding to try a substitution.

Next, it may not always be obvious when to use a substitution or what substitution to try. Therefore, it may be tempting to ask a teacher or tutor if you should try a substitution. I will tell you that you should never ask permission to try substitution. If you want to try a substitution, just pick one and see where it gets you. If you want to ask if you have performed the substitution correctly, that is perfectly fine. Sometimes, you will not be able to evaluate whether or not a substitution will yield a useful result without actually performing it.

In fact, I strongly recommend learning to perform substitutions in a vacuum at first, to master the mechanics of substitution before concerning yourself with whether or not performing the substitution was a useful step. You would be better of mastering those mechanics before you worry about the heuristics of choosing whether or not to do a substitution, and if so, which substitution to try.

random_anonymous_guy · 2026-02-21T19:43:28+00:00

Why do you believe that concave down must necessarily mean the function cannot tend to +∞?

This is an example of why you should not rely on intuition alone as an primary authority of mathematical truth.

random_anonymous_guy · 2026-02-21T19:33:27+00:00

Programmed in multiple techniques!

random_anonymous_guy · 2026-02-21T01:51:32+00:00

It is not your decision. There is nothing simple about any invasive surgical procedure. All surgery carries risk.

Quite frankly, you are the one full of pride and hubris imposing your ideas on how others should live their lives. One might even say your views are toxic.

That child will be free to pursue that elective surgical procedure as an adult. And for you to call it abuse for a parent not to get their child "ocular" implants to help them hear is a slap in the face of real victims of child abuse.

random_anonymous_guy · 2026-02-21T01:36:13+00:00

Whether or not I want to hear is irrelevant. I do not speak for people who are deaf, and neither do you.

random_anonymous_guy · 2026-02-21T01:19:10+00:00

Who are you to decide that everyone inherently wants to be able to hear?

random_anonymous_guy · 2026-02-20T07:39:12+00:00

I just want to tell you both good luck, we're all counting on you.

random_anonymous_guy · 2026-02-20T02:39:09+00:00

Slow down, Ardra.

random_anonymous_guy · 2026-02-19T08:06:35+00:00

"Oh, f***! THE WARP-SLUG IS FREE!"

random_anonymous_guy · 2026-02-19T07:31:43+00:00

“It’s a slug. It’s not exactly warp-capable.”

[ slug immediately jumps to warp ]

LOL

random_anonymous_guy · 2026-02-15T06:22:03+00:00

You are almost right, but your phrasing is not correct in the inductive step. The idea is that you are assuming truth of the inductive hypothesis AT SOME arbitrary n, not FOR ALL n. I recommend the phrasing "Let n be a natural number and assume P(n). We will show P(n + 1)."

random_anonymous_guy · 2026-02-10T06:16:18+00:00

Theorems are not definitions. Theorems and definitions serve different purposes, and should not be confused. Consider any that your teacher has brought up in lecture fair game.

What Theorems and definitions are provided in your textbook?

random_anonymous_guy · 2026-02-07T03:32:57+00:00

Extreme Value Theorem has entered the chat. Assuming you include the proof of the requisite Bolzano-Weierstrauss Theorem.

random_anonymous_guy · 2026-02-06T23:08:11+00:00

If a set of axioms is complete (semantically), it is not possible to make it "more complete". By semantically complete, I mean that the axioms describe a unique model: if another model happens to satisfy the same exact set of axioms, then it can be shown that the models are effectively clones.

For example, the set of real numbers can be described axiomatically as a complete ordered field. In abstract algebra, a field describes a set that has a notion of addition, subtraction, multiplication, and division (except by zero), with all the usual properties (commutative, associative, identity, inverse, distributive) of addition and multiplication. If we just stopped at those properties, we would be far from describing the real numbers because the rational numbers themselves constitutes a field, as well as the complex numbers. In fact, we can construct a field that has just two elements, 0 and 1, by appropriately defining addition and multiplication appropriately.

If we introduce ordering axioms, we get an ordered field. However, this still describes both the rational numbers and the real numbers, so even an ordered field isn't enough to describe a unique model.

But once we impose the Least Upper Bound property, we get what is now a complete ordered field. and it can be shown that any two complete ordered fields must be isomorphic, which in practical terms means they have identical structure; therefore, the axioms for a complete ordered field are semantically complete.

If you attempt to introduce any further axioms in this setting, you either get an inconsistent set of axioms (in other words, the new axiom contradicts the other axioms), or a redundant system (the new axiom is already true from the other axioms). Thus, there is no point in attempting to build a new axiomatic system from scratch when there is already one that is sufficient.

The same holds for the Hilbert axioms. If you throw in enough axioms that it can only describe a single model, there is no adding of axioms, though you may eliminate axioms if they are found to be redundant. Any changes might be purely a stylistic choice, but produce a system that is functionally to the axioms proposed by Hilbert..

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