The Natural Base e

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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
2.
If $f(x) = e^x$ , what is $f(0)$ ?
1. a) $0$
2. b) $1$
3. c) $e$
4. d) $-1$
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 $y = 3^x$
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
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) $9$
6.
Compute $\left(1 + \frac{1}{n}\right)^n$ for $n = 2$ .
7.
The graph of $y = e^x$ crosses the y-axis at the point $(0, b)$ . What is $b$ ?
8.
Using $e^2 \approx 7.389$ , evaluate $3e^2$ . Round to the nearest tenth.
9.
Using $e \approx 2.718$ , evaluate $e^{-1} = \frac{1}{e}$ . Round to three decimal places.
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.
11.
Compute $\left(1 + \frac{1}{n}\right)^n$ for $n = 4$ . Round the final answer to four decimal places.
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 $e^2 \approx 7.389$ .
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?
14.
1000 dollars is invested at 10% per year for one year, compounded semi-annually ( $n = 2$ ). Compare the result with continuous compounding, using $e^{0.1} \approx 1.10517$ .
15.
Evaluate $4e^{0.5}$ with the calculator, writing $e$ as $2.71828$ . Round the final answer to the nearest hundredth.
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.
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.
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 $e^{0.12} \approx 1.1275$ .