Mathematics 100 – Practice Midterm Problems
October, 2017
1. Determine analytically and graphically the solution set of the inequality
x 1 2 x 1.
x
Determine the domain of the function f x e x 1 ln arcsin e .
3
x
2.
x x 1
2
x
3. Consider the function given by f x ln .
x 1
a) What is the domain of f ?
b) Given that f is one-to-one on its domain, find a formula for the inverse
function f 1 x . What is its domain?
c) What are the ranges of f x and f 1 x ?
e x , x 1
4. Let f x .
x, x 1
Given that f is one-to-one on its domain, determine f 1 x and its domain.
5. a) Determine whether the sequences an converges or diverges. If it converges,
find the limit. Note that L’Hospital’s Rule is not permitted.
i) an 1
n n
n 1
4
ii) an 3 n 3 n 2 1 3 n 3 1
3 6n 3n cos n
8 2
6. a) Determine whether the sequence converges or diverges.
4
n 9
5n 3
sin n
b) Let a n be a sequence where a1 2, an1
1
, n 1.
3 an
i) Use induction to prove that 0 an 2.
ii) Use induction to prove that a n is monotonic decreasing.
iii) Find the limit of this sequence.
1
, 7. Given the recursively defined sequence
x1 1, xn1 15 2 xn , n 1.
a) Use induction to show that the sequence is (i) bounded above by 5, and (ii)
increasing.
b) Explain why you know the sequence converges.
c) Find the limit of the sequence.
8. Use induction to prove that
1 3 5 2n 1 n 2
for all positive integers n.
In other words, show that the sum of all odd positive integers from 1 to 2n 1 is
n2.
If f 0 x and f n1 f 0 f n for n 0,1,2, , find an expression for f n x
1
9.
2 x
and use mathematical induction to prove it.
10. Evaluate the following limits (no L’Hospital’s Rule!!):
a) lim
tan 1 x 2
x 1 x4 1
x
b) lim x 1 cos 2
x 1
x 1
c) lim x 1 1 x 1
2
x 1
2x 6
d) lim
x 3
9 x2
e) lim
2 x 2
3x 1 3/ 2
x 4x 3 x
cos x
f) lim
x e x
x
g) lim tan
x 1 2
h) lim
cos x100
x x
2
October, 2017
1. Determine analytically and graphically the solution set of the inequality
x 1 2 x 1.
x
Determine the domain of the function f x e x 1 ln arcsin e .
3
x
2.
x x 1
2
x
3. Consider the function given by f x ln .
x 1
a) What is the domain of f ?
b) Given that f is one-to-one on its domain, find a formula for the inverse
function f 1 x . What is its domain?
c) What are the ranges of f x and f 1 x ?
e x , x 1
4. Let f x .
x, x 1
Given that f is one-to-one on its domain, determine f 1 x and its domain.
5. a) Determine whether the sequences an converges or diverges. If it converges,
find the limit. Note that L’Hospital’s Rule is not permitted.
i) an 1
n n
n 1
4
ii) an 3 n 3 n 2 1 3 n 3 1
3 6n 3n cos n
8 2
6. a) Determine whether the sequence converges or diverges.
4
n 9
5n 3
sin n
b) Let a n be a sequence where a1 2, an1
1
, n 1.
3 an
i) Use induction to prove that 0 an 2.
ii) Use induction to prove that a n is monotonic decreasing.
iii) Find the limit of this sequence.
1
, 7. Given the recursively defined sequence
x1 1, xn1 15 2 xn , n 1.
a) Use induction to show that the sequence is (i) bounded above by 5, and (ii)
increasing.
b) Explain why you know the sequence converges.
c) Find the limit of the sequence.
8. Use induction to prove that
1 3 5 2n 1 n 2
for all positive integers n.
In other words, show that the sum of all odd positive integers from 1 to 2n 1 is
n2.
If f 0 x and f n1 f 0 f n for n 0,1,2, , find an expression for f n x
1
9.
2 x
and use mathematical induction to prove it.
10. Evaluate the following limits (no L’Hospital’s Rule!!):
a) lim
tan 1 x 2
x 1 x4 1
x
b) lim x 1 cos 2
x 1
x 1
c) lim x 1 1 x 1
2
x 1
2x 6
d) lim
x 3
9 x2
e) lim
2 x 2
3x 1 3/ 2
x 4x 3 x
cos x
f) lim
x e x
x
g) lim tan
x 1 2
h) lim
cos x100
x x
2
