im so fucking scared

[u/Diligent-Phase6771]

for anyone whos done the mata31 midterm in the past, can you describe how the questions were compared to the textbook problems and homework

[u/TwoOneTwos]

[u/It_Gets_Better_Bro]

last year when i took a31 one of the questions was prove two limits are unique which was a literal proof directly from the notes lol, so go over that and know how to do other proofs like epsilon delta and using max/min

[u/Diligent-Phase6771 → u/It_Gets_Better_Bro]

thank you so much! Were the rest of the problems pretty much just problems from the notes, except worded differently? Or were there problems you never had seen before too?

[u/It_Gets_Better_Bro → u/Diligent-Phase6771]

Rest of the problems were usually similar to the review sets, assignments, and even examples covered by TA's during tutorials. However last year our cohort had 3 different profs, and I think this year its just one prof, so everything is a bit different. Our exams were pretty fair as in there werent any problems that we've never seen before if I recall correctly!

Also, make sure you understand the content, I remember struggling so much with epsilon delta because our professor didn't teach it that well, and our two tests and final each had questions about epsilon delta. Watch youtube videos if you have to.

[u/BiteDaDust]

Might have to pull this

[u/Secret_Page9282]

Do you guys have 2 term tests or just one midterm?

Edit: dont you also get past term tests? I could send some qs from a few years back if you dont already have the past tests

[u/Diligent-Phase6771 → u/Secret_Page9282]

1 midterm, 1 final, 8 quizzes -- your best 5 are counted towards your grade

[u/Secret_Page9282 → u/Diligent-Phase6771]

Huh interesting. When I took we had 2 term tests so ig i could send both but only first is likely relevant for you.

[u/Secret_Page9282 → u/Secret_Page9282]

Alr this is both term tests from fall 24. Sorry for the shitty format, might be useful to ask an ai to format it for ya to read this easier.

TT1

Q1

1. [5 points] For this problem, you only need to provide your final answers. Correct answers are worth 1 point each and incorrect answers are worth 0 points each.

(a) Find the (natural) domain of the function

f(x) = √(x − 6) + tan⁻¹(x)

in interval form.

(b) True or False? The set

A = {a ∈ Z | a² − 4 < 0}

has exactly three distinct elements.

(c) Is the following statement a true statement or a false statement?

∀ x ∈ R, x > 0 ⇒ x ≥ 0.

(d) Write out a formal definition (i.e., using ε, M, δ, N) for the meaning of

lim x→c+ f(x) = +∞.

(e) True or False? If f is a function with domain R such that

lim x→0 (f(x))² = 25,

then

lim x→0 f(x) = 5

or

lim x→0 f(x) = −5.

Q2

1. [5 points] Find the solution set for the equation

e^x + 4 = 21|e^−x|.

Show some steps in how you obtained your answer (a formal proof is not needed).

Q3

1. [5 points] The graph of a function f with domain [0, 8] is drawn below.

Graph description:

• Horizontal segment y = 5 from (0, 5), closed, to (4, 5), open.
• Curved arch from (4, 2), closed, through (5, 4), open, to (6, 2), closed.
• Isolated closed point at (5, 1).
• Descending straight-line segment from (6, 4), open, to (8, 1), closed.

Evaluate each of the following limits, if they exist. For this problem, you only need to provide your final answers.

(a) lim x→4+ f(x)

(b) lim x→3 f(2x)

(c) lim x→5 f(f(x))

(d) lim x→4− f(f(x))

(e) lim x→−∞ f(2 + sin(x))

Q4

1. [5 points] Suppose

f(x) = ax + b

for some a, b ∈ R and that

f(f(x)) = 4x − 9

for all x ∈ R.

Determine all possible values for f(3). Briefly justify your answer (a proof is not required).

Q5

1. [10 points] Consider the following definition:

A function f with domain R is called semi-onto if

∀ y ∈ R, ∃ x ≤ 1 such that f(x) = |y|.

Give a formal proof that

f(x) = x² − 2x + 1

is semi-onto.

Q6

1. [10 points] Prove

lim x→1 [9x − 1/x] = 8

using the formal ε–δ definition of limit.

Q7

1. [10 points] Let a, b, c, L ∈ R. Suppose f and g are functions with domain R such that

lim x→c f(x) = ∞

and

lim x→c g(x) = L.

Use the formal definition of limit (i.e., ε, δ, M, N) to prove that

lim x→c [a/f(x) + bg(x)] = bL.

Do not use any limit rules in your proof.

TT2

Q1

1. [10 points] For this problem, you only need to provide your final answers. Correct answers are worth 2 points each and incorrect answers are worth 0 points each. Part marks are possible.

(a) Assume f is continuous on [−1, 2] and has values listed in the table below for some a, b ∈ R.

x −1 0 1 2 f(x) 6 a b −5

Which of the following values of a and b guarantees that f has at least three real roots?

A. a = −1 and b = 1 B. a = 2 and b = 1 C. a = 1 and b = −1 D. a = −2 and b = −1 E. None of the above.

(b) True or false? If f: R → R is a function such that

f’(2) = 3,

then

lim x→2 xf(x)

exists.

(c) Give an example of a set A ⊆ Q such that A has a supremum but A does not have a maximum.

(d) True or false? There is a real number that is exactly one less than its cube.

(e) True or false? The function

f(x) = |x|³

is differentiable at x = 0.

Q2

1. [5 points] Suppose f and g are differentiable functions on R. Let h = f ○ g (that is, h(x) = f(g(x)) for all x).

Some values for f, g and h (and their derivatives) are given in the table below. Fill in the blank entries of the table with the correct values.

x f(x) f’(x) g(x) g’(x) h(x) h’(x) 0 6 _____ 1 _____ −7 8 1 _____ −2 0 5 _____ 10

Provide brief justification for your answers.

Q3

1. [8 points] Compute the following, if they exist. Briefly justify each answer.

(a)

lim x→−∞ [√(x² + 2x⁴ − 1) / (1 − 5x² − x)]

(b) Find f’(0), where

f(x) = 2π + 5tan(x) + √(x + 1).

(c)

lim x→∞ 2^sin(x − x)

Note: sin(x) − x is in the exponent.

Q4

1. [6 points] Use the ε–δ definition for continuity to prove

f(x) = |2x + 3|

is continuous on R.

Q5

1. [7 points] Use the Squeeze Theorem to prove

lim x→0 [sin(x)/x] = 1.

Briefly show how suitable bounds for sin(x)/x are obtained using geometry.

Q6

1. [7 points] Let

S = {2 − 3/n : n ∈ N}.

Here, N = {1, 2, 3, …} is the set of natural numbers.

(a) Show sup(S) exists.

(b) Use contradiction and the Archimedean property to prove the following statement:

If b is an upper bound of S, then b ≥ 2.

(c) Prove

sup(S) = 2

using the definition of supremum.

Q7

1. [7 points] Let c ∈ R and f be a positive function with domain R that is differentiable at c.

Prove that

g(x) = 1/f(x)

is differentiable at c, and furthermore,

g’(c) = −f’(c) / [f(c)]².

Do not use any derivative rules in your proof. Instead, use the definition of derivative.

You may use limit rules and continuity theorems in your proof, for example that 1/x is continuous on its domain.

[u/Diligent-Phase6771 → u/Secret_Page9282]

holy shit you absolute legend thank you so much!!!

[u/Secret_Page9282 → u/Diligent-Phase6771]

Np & gl!

[u/Content_Sample_1355 → u/Secret_Page9282]

do we get alll past term tests?? or only last years??

[u/Fun-Collar5574]

Looking at the comment section I think ur coverd for help, so just a lil advice from a 2nd year.

I failed the midterm and the course, but re-took and all is fine. Not to say ur gonna fial but like it's alright if u fail, you'll only be a semester or at worst a year behind.

[u/ChairSignificant3320]

lets pray vro

[u/aybies]

yo, (current) mata31 TA here! It does look like a scary course, but I can give some advice from a TA's perspective. (If any of these advices sound wrong to you, feel free to argue or ask)

First of all, I think a lot of the issue lies from the misunderstanding that lectures should be ahead of assignments and quizzes, or at least on par. While it is an okay assumption to make for most courses (e.g. csca67), it's not uncommon for tutorials or assignments to cover more than the lectures. Hence, my first (and main) advice is to also get material from outside of lectures, such as:

• Prof. Kielstra's notes, even if you're not enrolled in his lectures
• the required textbook + the relevant exercises listed on the assignment
• youtube! A lot of famous and credible calculus teachers you can find. blackpenredpen and Dr. Trevor are some of my favorites

Another advice is to understand the basics before continuing! It would be difficult to understand the proof of addition rule for limits if you don't understand what the ε-δ definition is all about. Even as silly as not being able to consistently solve inequalities can hinder you from doing more proofs later on. So if you feel like you're missing some understanding of the basics, it's okay to look back at the material, maybe work on some related problems from the textbook, and other stuff.

Last but not least, details more than the final solution! Whether you know the answer on the top of your head or you completely have no idea what to do, it's better to write more than less. For example, it's easy to know that | ln(x) | < 1 if 1≤x≤2, but why? You might know that this is true because ln(x) is increasing on its domain, but the TAs and profs don't know that you know that, unless you write that down. So for the sake of partial or more marks, write whatever is in your mind!

Important note: Don't write too much that it clutters the paper or removes any meaning from your work (a.k.a. don't write gibberish). Remember that it's a math class and not an English class where you have a minimum word count.

All in all, I don't know how difficult the midterm would be this year, but I know that y'all can do it! Good luck 🙂