Lesson test: Exponentials and growth

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Exponentials and growth

1.[2p]
Why is $b^{0}$ equal to $1$ rather than $0$ ?

It is the only value that keeps $b^{m} \times b^{n} = b^{m + n}$ true

Correct

It is a convention with no reason behind it

The answer is: It is the only value that keeps $b^m \times b^n = b^{m+n}$ true

Because zero factors multiply to zero

The answer is: It is the only value that keeps $b^m \times b^n = b^{m+n}$ true

Because $b$ is always positive

The answer is: It is the only value that keeps $b^m \times b^n = b^{m+n}$ true

2.[2p]
Evaluate $27^{4 / 3}$ .

CorrectNot quite: 81

3.[2p]
Evaluate $16^{- 3 / 4}$ .

CorrectNot quite: 0.125

4.[3p]
A balance of 1000 pounds earns 6 per cent a year compounded monthly. What is the balance after five years, in pounds?

CorrectNot quite: 1348.85

5.[2p]
To three decimal places, what is the limit of $(1 + 1 / n)^{n}$ as $n$ grows without bound?

CorrectNot quite: 2.718

6.[3p]
Carbon-14 has a half-life of 5730 years. What fraction of the original remains after 15,000 years?

CorrectNot quite: 0.163

7.[2p]
A polynomial of high enough degree eventually overtakes $2^{x}$ and stays ahead.

True

The answer is: False

False

Correct

8.[3p]
Why does the definition of $2^{\sqrt{2}}$ need the completeness of the real line?

It is defined as the value that rational powers approach, and completeness guarantees such a value exists

Correct

Because $\sqrt{2}$ is negative

The answer is: It is defined as the value that rational powers approach, and completeness guarantees such a value exists

Because the index laws fail for irrational exponents

The answer is: It is defined as the value that rational powers approach, and completeness guarantees such a value exists

Because $2^{\sqrt{2}}$ is itself rational

The answer is: It is defined as the value that rational powers approach, and completeness guarantees such a value exists

9.[2p]
Match each base to the behaviour of $y = b^{x}$ .

$b = 2$
$b = 0.5$
$b = 1$
$b = e$

increases at a rate equal to its own value
decays toward zero
increases without limit
stays constant

Show the answer

$b = 2$ : increases without limit $b = 0.5$ : decays toward zero $b = 1$ : stays constant $b = e$ : increases at a rate equal to its own value

1.[2p] Why is b0 equal to 1 rather than 0?

2.[2p] Evaluate 274/3.

3.[2p] Evaluate 16-3/4.

4.[3p] A balance of 1000 pounds earns 6 per cent a year compounded monthly. What is the balance after five years, in pounds?

5.[2p] To three decimal places, what is the limit of (1+1/n)n as n grows without bound?

6.[3p] Carbon-14 has a half-life of 5730 years. What fraction of the original remains after 15,000 years?

7.[2p] A polynomial of high enough degree eventually overtakes 2x and stays ahead.

8.[3p] Why does the definition of 22 need the completeness of the real line?

9.[2p] Match each base to the behaviour of y=bx.

1.[2p]
Why is $b^{0}$ equal to $1$ rather than $0$ ?

2.[2p]
Evaluate $27^{4 / 3}$ .

3.[2p]
Evaluate $16^{- 3 / 4}$ .

4.[3p]
A balance of 1000 pounds earns 6 per cent a year compounded monthly. What is the balance after five years, in pounds?

5.[2p]
To three decimal places, what is the limit of $(1 + 1 / n)^{n}$ as $n$ grows without bound?

6.[3p]
Carbon-14 has a half-life of 5730 years. What fraction of the original remains after 15,000 years?

7.[2p]
A polynomial of high enough degree eventually overtakes $2^{x}$ and stays ahead.

8.[3p]
Why does the definition of $2^{\sqrt{2}}$ need the completeness of the real line?

9.[2p]
Match each base to the behaviour of $y = b^{x}$ .