MATH 100 final review problems:
1. (2010)√Compute the linear approximation of f (x) =
estimate 4.16.
√
4 + x at x = 0 and use it to
2. (2010) What is the second-degree Taylor polynomial of f (x) = ecos(x) about x = π2 .
3. (2022, midterm 3) Fully simplify ln(cosh2 (x) + sinh2 (x) + sinh(2x)).
You may use the definition of cosh and sinh, but you are not permitted to use other hyperbolic
identities without a proof.
4. (2021, midterm 3) Find the absolute maximum and minimum values of the function
f (x) = 23 (x2 − 1)2/3 − x2 . on the interval [−1, 3].
5. (2020, Written 3) Show that | arctan(x) − arctan(y)| ≤ |x − y|.
6. (2012) Show that the equation f (x) = x4 − 3x + 1 has exactly 2 real roots.
7. (2015) How many solutions does the equation ex = e−x have?
8. (2022) Find the slant asymptotes of the graph of the function f (x) = x arctan(x).
9. (2021) The function f is given by
f (x) =
Its first and second derivatives are f ′ (x) =
x3
.
x3 + 2
6x2
−24x(x3 − 1)
′′
and
f
(x)
=
.
(x3 + 2)2
(x3 + 2)3
a) Find the domain of f (in interval form) and all asymptotes of f .
b) Find the intervals on which f increases and the intervals on which it decreases. Find all
local maximum and minimum values of f .
1
c) Find the intervals on which f is concave upward and concave downward. Find the inflection points.
d)Use the results of parts (a-c) to sketch the graph of f .
10. Evaluate the following limits or explain why it does not exist:
i. (2002) lim+ sin(x) ln(x)
x→0
ii. (2021) lim Z
(arctan(x))2
x→0
1−x
sin(πt2 )dt
1
n
3
1 X ni 3
iii. (2021) lim
4
n→∞ n
1 + ni 4
i=1
11. (2021) A cone is obtained by rotating a right triangle around one of its legs; see the
figure.
√ Find the largest possible volume of 1such2 a cone, if the hypotenuse of the triangle is
10 3 cm. The volume of a cone is Vcone = 3 πr h.
12. (Winter 2015) Find the point on the parabola y 2 = 2x that is closest to the point (1, 4).
13. (2018) Rewrite the following sum as a definite integral lim
n→∞
Z
14. (2016) Evaluate
∞
X
i=1
sin
1 + cos
πi
n
2 ·
πi
n
π
.
2
3
(4 − 3x2 )dx by using the definition of a definite integral.
1
Z
x
15. (2022) Let f (x) =
√
3 + et dt. Find the third degree Taylor polynomial for the
0
function f centered at a = 0.
2
16. (Winter 2016) Find a function f and a number a such that:
Z
x
6+
a
√
f (t)
dt = 2 x.
2
t
17. Evaluate the integrals
1
Z
√
i)(2002)
−1
2
dx
4 − x2
2
Z
(2x)(x2 − 2)2/3 sin(x2 − 2)dx
ii)(2020)
0
1
Z
iii)(2014)
−1
x2
6
√
Z
iv)(2017)
1
dx
+ 4|x| + 4
36 − x2 dx
−6
Z
4
||2x − 6| − x| dx
v) (2021)
0
3