Functions and Sequences (Chapter 2 & 3)

Question 1

Identify the range of the following function

Answer 1

{-5, -π} ∪ (0, 7)

Question 2

True or false?

(f∘f^-1)(x) = (f^-1∘f)(x)

Answer 2

True.

It is commutative in this case. However, note it is not always commutative like (f∘g)(x) may not be equal to (g∘f)(x) when g and f are different functions.

Question 3

Sketch the inverse of a sin function

Answer 3

Initially, you cannot get an inverse of a sin as it is not one-to-one. But when you restrict the domain, it is possible which gives you the arcsin(x) function

Question 4

Sketch the inverse of a cosine function

Answer 4

Initially, you cannot get an inverse of a cos as it is not one-to-one. But when you restrict the domain, it is possible which gives you the arccos(x) function

Question 5

Sketch the inverse of the tan function

Answer 5

Question 6

What is tan(pi/4)

Answer 6

1

Question 7

What is tan (pi/6)

Answer 7

1/sqrt(3) = sqrt(3)/3

Question 8

What is tan (pi/3)

Answer 8

sqrt(3)

Question 9

Given a sequence {a_n}⁷_n=0

When does the sequence start and end?

Answer 9

Starts at a₀ and ends at a₇

Question 10

What can be said about convergent sequences?

Answer 10

Convergent sequences have a limit.

It can be said that the sequence will ‘converge’ to the limit

Question 11

Determine if the following sequence is convergent or divergent?

Answer 11

Question 12

True or false?

Since both cos(nπ) and sin(nπ) alternate, they are both divergent.

Answer 12

False

Question 13

True or false?

Adding two limits that are divergent will always give a divergent limit.

Answer 13

False.

Question 14

Identify if the following sequence converges or not.

{a_n}=rⁿ

for |r|<1

Answer 14

Converges to 0
|r|<1 basically means -1<r<1.
And so it will become some fraction of 1 (1/2, -1/3, 1/4, etc.). Raising these numbers to higher powers will make them become closer to zero whether they are negative or positive (the denominator will get larger and larger)

Question 15

Identify if the following sequence converges or not.

{a_n}=1ⁿ

Answer 15

Converges to 1

Question 16

Identify if the following sequence converges or not.

{a_n}=(-1)ⁿ

Answer 16

Divergent

{a_n}=1,-1, 1,-1, 1,-1, 1,-1…

Question 17

Identify if the following sequence converges or not.

{a_n}=rⁿ

for |r|>1

Answer 17

Sequence is divergent
For any number larger than 1, a_n will get larger as it will be raised to larger powers (either negative way or postive way). And so the limit would be infinity meaning it D.N.E

Question 18

Evaluate the following limit

Answer 18

lim=1. Converges to 1

Notice the 1/n exponential. So as n gets larger, exponent will get closer to 0. And so c^1/n will get closer to 1 (c⁰=1)

Question 19

Evaluate the following limit

for r>0

Answer 19

Converges to 0. Note that as n gets larger, the demonimator (n^r) will aproach a large number. And 1 divided by a larger and larger number will become closer and closer to 0

Question 20

True or false?

The following limit is known as an indeterminate

Answer 20

False.

Recall that an indeterminate is when there is ‘competition’ where the number wants to get larger and smaller at the same time such as: (aⁿ)/n

Question 21

Evaluate the following limit

Answer 21

1

Note that the base, n gets larger but the index, 1/n gets smaller. And so there is a competition going on. So it is either the limit aproaches infinity (base wins the competition) or the limit aproaches 1 (index wins the competition, 1/n aproaches 0). You just have to remember that the index wins

Question 22

True or false.

The following is an example of an indeterminate

Answer 22

True.

There is a ‘competition’ going on. The base, n gets larger but the index, 1/n gets smaller

Question 23

Evalutate the following limit

Answer 23

0

As n gets larger, the denominator will get larger, and the number will get smaller and smaller (aproaching 0)

Question 24

Evaluate the following limit

Answer 24

0

This is an example of an indeterminate. The numerator will try to make the limit to get larger and larger as n increases while the denominator will try to make the limit smaller as n increases. The question is, who will win. In this case, factorials have a greater rate of change (on average) and so the denominator will win and cause the limit to aproach 0.

Question 25

# Evaluate the following limit

Answer 25

e ## Footnote As n gets larger and larger, the number inside the bracket will aproach 1. This is because 1/n will aproach 0; adding 1 to that will equal 1. Raising a number close to 1 by n (for any n number) will not be 1 because the base is not exactly 1. It will try to make it large. And so the question is, what number does it reach? The fact is it reaches e, eulers number.

Question 26

# True or false? The following is an example of an indeterminate

Answer 26

True ## Footnote There is a competition going on. As n gets larger and larger, the number inside the bracket will aproach 1. This is because 1/n will aproach 0; adding 1 to that will equal 1. Raising a number close to 1 by n (for any n number) will not be 1 because the base is not exactly 1. It will try to make it large.

Question 27

Explain the meaning behind euler's number *e*

Answer 27

Euler's number is actually obtained from the following limit ## Footnote Basically, when you have compound interest at a 100% interest rate, As you compound it for longer and longer, your amount will slowly be multiplied by Euler's number each time

Question 28

# Evaluate the following limit

Answer 28

e^a ## Footnote Just have to remember this. Note the simularity between this and euler's limit

Question 29

# Given sequences: {a_n}, {b_n}, {c_n} Explain the squeeze theorum

Answer 29

Question 30

Answer 30