Math 156
Applied Honors Calculus II
Fall 2025
hw1 , due: Tuesday, September 2 at 4pm
Write the solutions neatly; number each page; scan them into a PDF using a scanning app (if you
need help with that, ask me); don’t take a photograph (the background comes out too dark and
the writing is hard to read); upload the PDF into Gradescope on the Canvas site.
0. (optional) Describe your academic background and interests. What year are you in? Have you
decided on your major? How did you hear about this class? What other STEM classes are you
taking this semester?
1. Write the sum in sigma notation.
a) 1 + 3 + 5 + 7 + · · · + (2n − 1)
b) 1 + 2 + 4 + 8 + 16 + 32
c) 1 +
1
1
1
1 1
+ +
+
+
4 9 16 25 36
d) x + x2 + x3 + · · · + xn
e) 1 − x + x2 − x3 + · · · + (−1)n xn
2. True or False? Give a reason or counterexample to justify your answer.
a) (f (x)g(x))0 = f 0 (x)g 0 (x)
b)
n
X
n
X
i=0
i=0
(n − i) =
i
c)
5
X
((i + 1)3 − i3 ) = 215
i=1
d)
a c
a+c
+ =
b d
b+d
3. Prove the following results. Follow the similar examples we did in class.
n
n
n
n
X
X
X
X
n(n + 1)(2n + 1)
i2 =
bi b)
ai +
(ai + bi ) =
a)
(hint: start from (i + 1)3 − i3 = . . .)
6
i=1
i=1
i=1
i=1
4. Each set below defines a region in the xy-plane. Sketch the region, express the area as a limit
of Riemann sums, and evaluate the limit.
a) {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 + x}
b) {(x, y) : −1 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x2 }
c) {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ ex }
note: problem (c) requires the formula for the sum of a finite geometric series; if you know it, then
go ahead and evaluate the limit; otherwise skip this problem for now; you will learn the formula
on hw2
n X
2i 2 2
5. The expression lim
1+
is a limit of Riemann sums giving the area of a region in
n→∞
n n
i=1
the xy-plane defined by {(x, y) : a ≤ x ≤ b, 0 ≤ y ≤ f (x)}. Assuming a = 1, find b and f (x);
sketch the region; evaluate the limit. (hint: what is ∆x? what is xi ?)
optional: check the limit you computed above using the Fundamental Theorem of Calculus; if you
don’t know how to do this, then skip it for now; it will be discussed soon in class
6. Sketch the graph of the function on the interval 0 ≤ x ≤ 2π. Label the axes.
a) f (x) = sin x
b) f (x) = sin 2x
c) f (x) = sin2 x