MAT 137
Tutorial #13– Anti-derivatives and Integral as limits
January 20/21, 2026
Due on Friday, Jan 23 by 11:59pm via GradeScope
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Learning outcomes of this Tutorial:
You should be able to
• Use the definition to compute lower and upper sums.
• prove a function is integrable.
• compute the general anti-derivative of a given function.
• prove the lower integral exists.
Criteria:
The tutorial is worth 2 points. The rubric is below.
Proficient
Emerging
Needs Improvement
Algebraic accuracy
Includes most steps and
steps are accurate (1)
missing a few steps
and/or some steps are
inaccurate (0.5)
many
missing
steps
and/or many inaccurate
steps (0)
Mathematical
language
reasoning is correct and
mostly correct math
proof language (1)
reasoning is mostly correct and mostly correct
mathematics proof language (0.5)
Much of the reasoning
and language is incorrect
(0)
1. Warm-up: Determine if the following statements are true or false. Circle your answer.
(a) sup{1, 2} = sup{x : 1 < x < 2}.
True
False
(b) max{1, 2} = max{x : 1 < x < 2}.
True
False
(c) inf{1, 2} = inf{x : 1 < x < 2}.
True
False
(d) min{1, 2} = min{x : 1 < x < 2}.
True
False
2. Consider a function f : [a, b] → R which is increasing on [a, b].
(a) Let P be a partition of [a, b] such that all n subintervals have the equal length. Show that
UP (f ) − LP (f ) = (f (b) − f (a)) · b−a
.
n
(b) Show that f (x) is integrable on [a, b]. You may use the following theorem in your proof: A
function f is integrable if
∀ > 0, ∃ a partition P such that UP (f ) − LP (f ) < .
3. Consider the function on [0, 1]
(
x x ∈ Q, 0 ≤ x ≤ 1
f (x) =
0 x∈
/ Q, 0 ≤ x ≤ 1
(a) Given any partition P = {0 = x0 , x1 , · · · , xn = 1} of [0, 1], find the LP (f ).
(b) What is I 10 (f )?
1
(c) Given any partition P = {x0 , x1 , · · · , xn } of [0, 1], prove that UP (f ) > .
2
xi−1 + xi
since xi > xi−1 . You may
Hint: Note that Mi = sup of f on [xi−1 , xi ] = xi and xi >
2
need this formula (a + b)(a − b) = a2 − b2 .
(d) Is this function integrable on [0, 1]? Explain your answer.
More Practice Problems:
4. Let a < b. Let f be a bounded function on [a, b]. As you know, we define the lower integral of f on
[a, b] as a supremum, specifically:
Iab (f ) = sup {LP (f )| P is a partition of [a, b]}
This is well-defined. In other words, this set must necessarily have a supremum. Prove it.
Note: In the videos we skipped this detail. We just stated that the supremum must necessarily exist
and we left it as an exercise for you to justify why.
Hint: Define A = {LP (f )| P is a partition of [a, b]}. Prove that A is bounded above and non-empty.
Then, by the least-upper-bound property, it must have a supremum.
5. Find the antiderivatives.
Z 3
x + 2x2
(a)
dx
x
Z
2
(b)
dx
3x − 1
Z
(c)
tan2 x dx Hint: Think of the trig identity involving tangents and secants.
Z
√
x (x + 1) dx
(d)
Z
100
(e)
x 3x2 + 1
dx
6. Next, try some harder antiderivatives. Remember: the key is often to make an educated guess, try
it, and then take it from there.
Z
Z
Z
10
(3x + 7) dx
3 sin(2x) dx
5e−2x dx
Z
Z
1
2
√
dx
dx
3
4
(7 − 6x)
5 − 2x
7. These antiderivatives are more difficult. We will learn later a more systematic way to approach them,
but it is a very good exercise to try to figure them out now without knowing any tricks or formulas,
just by (sophisticated) guess and check. You will solidify your understanding in the process.
√
Z
Z
cos x
2
√
dx
xe−x dx
x
Z
Z
Z
3
(ln x)
sin(ln x)
1
dx
dx
dx
x
x
x ln x
Hint: For the first one, study the derivative of a function of the form F (x) = (3x2 + 1)n for some
appropriate value of n.
√
For the second one, what is the derivative of F (x) = sin x ?