The Natural Base e

Show your work for each problem.

1.
Which number is the best approximation of $e$ ?
1. a)2.5
2. b)1.41421
3. c)3.14159
4. d)2.71828
Answer: 2.71828
$e \approx 2.71828$ . The number $3.14159$ is $\pi$ , $1.41421$ is $\sqrt{2}$ , and $2.5$ is too small.
2.
If $f(x) = e^x$ , what is $f(0)$ ?
1. a) $0$
2. b) $1$
3. c) $e$
4. d) $-1$
Answer: $1$
$f(0) = e^0 = 1$ . That is why the graph of $y = e^x$ crosses the y-axis at $(0, 1)$ .
3.
For $x > 0$ , where does the graph of $y = e^x$ lie?
1. a)Above $y = 2^x$ and below $y = 3^x$
2. b)Exactly on the graph of $y = 3^x$
3. c)Above both $y = 2^x$ and $y = 3^x$
4. d)Below both $y = 2^x$ and $y = 3^x$
Answer: Above $y = 2^x$ and below $y = 3^x$
Since $2 < e < 3$ , for $x > 0$ we get $2^x < e^x < 3^x$ . For example at $x = 2$ : $4 < 7.389 < 9$ .
4.
What happens to $\left(1 + \frac{1}{n}\right)^n$ as $n$ gets larger and larger?
1. a)It gets closer and closer to $1$
2. b)It gets closer and closer to $e \approx 2.71828$
3. c)It gets closer and closer to $2$
4. d)It grows without limit
Answer: It gets closer and closer to $e \approx 2.71828$
The values are $2$ , $2.25$ , $2.613$ , $2.7146$ , ... They keep increasing but level off at $e \approx 2.71828$ , so they never grow without limit and never go back to $1$ or $2$ .
5.
Using $e \approx 2.71828$ , which value is closest to $e^3$ ?
1. a) $27$
2. b) $20.09$
3. c) $8.15$
4. d) $9$
Answer: $20.09$
$e^3 \approx 2.71828^3 \approx 20.09$ . The value $8.15$ is $3e$ (a common mistake), and $9 = 3^2$ and $27 = 3^3$ use the wrong base. Since $e < 3$ , $e^3$ must be less than $27$ .
6.
Compute $\left(1 + \frac{1}{n}\right)^n$ for $n = 2$ .
Answer: 2.25
$\left(1 + \frac{1}{2}\right)^2 = 1.5^2 = 2.25$ . This is the second value in the table that approaches $e$ .
7.
The graph of $y = e^x$ crosses the y-axis at the point $(0, b)$ . What is $b$ ?
Answer: 1
At $x = 0$ : $y = e^0 = 1$ , so $b = 1$ . Every function $y = b^x$ passes through $(0, 1)$ .
8.
Using $e^2 \approx 7.389$ , evaluate $3e^2$ . Round to the nearest tenth.
Answer: 22.2
$3e^2 \approx 3 \cdot 7.389 = 22.167$ , which rounds to $22.2$ .
9.
Using $e \approx 2.718$ , evaluate $e^{-1} = \frac{1}{e}$ . Round to three decimal places.
Answer: 0.368
$e^{-1} = \frac{1}{e} \approx \frac{1}{2.718} \approx 0.36792$ , which rounds to $0.368$ .
10.
1000 dollars is placed in an account that pays 100% interest per year, compounded quarterly ( $n = 4$ ). How much is in the account after one year? Round to the nearest cent.
Answer: 2441.41
$1.25^4 = 2.44140625$ , so $A = 1000 \cdot 2.44140625 = 2441.40625 \approx 2441.41$ dollars. That is more than yearly compounding (2000 dollars) and less than the continuous limit $1000e \approx 2718.28$ dollars.
11.
Compute $\left(1 + \frac{1}{n}\right)^n$ for $n = 4$ . Round the final answer to four decimal places.
Answer: 2.4414
1. What is $1 + \frac{1}{4}$ as a decimal? 1.25
2. Square it: $1.25^2 = ?$ 1.5625
3. Square again: $1.5625^2$ , rounded to four decimal places 2.4414
12.
Let $f(x) = e^x$ . Find $f(0)$ , $f(1)$ and $f(2)$ , then check how the values grow. Use $e \approx 2.71828$ and $e^2 \approx 7.389$ .
Answer: 2.72
1. What is $f(0) = e^0$ ? 1
2. What is $f(1) = e^1$ , rounded to three decimal places? 2.718
3. What is $f(2) = e^2$ (use the given value)? 7.389
4. Divide $f(2)$ by $f(1)$ : $\frac{7.389}{2.718}$ , rounded to two decimal places 2.72
13.
Compare $2^2$ , $e^2$ and $3^2$ using $e^2 \approx 7.389$ . How far is $e^2$ from each of the other two values?
Answer: 1.611
1. What is $2^2$ ? 4
2. What is $3^2$ ? 9
3. How much bigger is $e^2$ than $2^2$ ? ( $7.389 - 4$ ) 3.389
4. How much smaller is $e^2$ than $3^2$ ? ( $9 - 7.389$ ) 1.611
14.
1000 dollars is invested at 10% per year for one year, compounded semi-annually ( $n = 2$ ). Compare the result with continuous compounding, using $e^{0.1} \approx 1.10517$ .
Answer: 2.67
1. What is the interest rate per half-year as a decimal? 0.05
2. What is the growth factor for one year, $1.05^2$ ? 1.1025
3. How much is in the account with semi-annual compounding? 1102.5
4. How much is there with continuous compounding? ( $1000 \cdot 1.10517$ , to the nearest cent) 1105.17
5. What is the difference between the two amounts? 2.67
15.
Evaluate $4e^{0.5}$ with the calculator, writing $e$ as $2.71828$ . Round the final answer to the nearest hundredth.
Answer: 6.59
1. What is $e^{0.5} = \sqrt{e}$ , rounded to four decimal places? (type 2.71828^0.5) 1.6487
2. Multiply by $4$ : $4 \cdot 1.6487 = ?$ 6.5948
3. Round $6.5948$ to the nearest hundredth 6.59
16.
1000 dollars earns 100% interest per year, compounded daily ( $n = 365$ ), for one year. How far below the continuous limit $1000e$ is the result? Use $e \approx 2.71828$ , round both amounts to the nearest cent and give the difference in dollars.
Answer: 3.71
1. Compute $\left(1 + \frac{1}{365}\right)^{365}$ , rounded to four decimal places 2.7146
2. How much is in the account after one year, to the nearest cent? 2714.57
3. What is the continuous limit $1000e$ , to the nearest cent? 2718.28
4. What is the difference between the two amounts? 3.71
17.
Show that at $x = -1$ the value of $e^x$ still lies between $2^x$ and $3^x$ . Use $e \approx 2.718$ and round to three decimal places.
Answer: 0.368
1. What is $2^{-1}$ ? 0.5
2. What is $3^{-1}$ , rounded to three decimal places? 0.333
3. What is $e^{-1} = \frac{1}{2.718}$ , rounded to three decimal places? It should land between your two answers. 0.368
18.
1000 dollars is invested at 12% per year for one year. How much more do you get with continuous compounding ( $A = Pe^{rt}$ ) than with yearly compounding? Use $e^{0.12} \approx 1.1275$ .
Answer: 7.5
1. What is the exponent $rt$ ? 0.12
2. What is the amount with continuous compounding? ( $1000 \cdot 1.1275$ ) 1127.5
3. What is the amount with yearly compounding? ( $1000 \cdot 1.12$ ) 1120
4. How much more does continuous compounding give? 7.5