4.05 The natural exponent
Use a calculator or other technology to approximate the following values correct to four decimal places:
e^{4}
e^{ - 1 }
e^{\frac{1}{5}}
5 \sqrt{e}
\dfrac{4}{e}
\dfrac{8}{9 e^{4}}
The natural base e, Euler's number, is defined as: e = \lim_{n \to \infty} \left(1 + \dfrac{1}{n}\right)^{n}
The table shows the values of \left(1 + \dfrac{1}{n}\right)^{n} for certain values of n:
| n | \left(1 + \dfrac{1}{n}\right)^{n} |
|---|---|
| 1 | 2 |
| 100 | 1.01^{100} = 2.704\,813 ... |
| 1000 | 1.001^{1000} = 2.716\,923 ... |
| 10\,000 | 1.0001^{10000} = 2.718\,145 ... |
| 100\,000 | 1.000\,01^{100000} = 2.718\,268 ... |
Evaluate \left(1 + \dfrac{1}{n}\right)^{n} for n = 1\,000\,000, correct to six decimal places.
Find a decimal approximation of e correct to nine decimal places.
It is possible to approximate e^{x} using the following formula:
e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \text{. . .} + \frac{x^{n}}{n!} + \text{. . .}
The more terms we use from the formula, the closer our approximation becomes.
Use the first five terms of the formula to estimate the value of e^{0.7}.
Round your answer to six decimal places.
Use the {e^{x}} key on your calculator to find the value of e^{0.7}.
Round your answer to six decimal places.
What is the difference between these two results?
Simplify the following expressions:
Expand and simplify the following expressions:
Fully factorise the following expressions:
The functions y = 2^{x}, y = e^{x} and y = 3^{x} have been sketched on the same axes:
For what values of x does the inequality 3^x > e^x hold?
For what values of x does the inequality 2^x > e^x hold?
Consider the function f \left( x \right) = e^{x}.
Complete the following table of values. Round each value to two decimal places.
| x | -5 | -3 | -1 | 0 | 1 | 3 | 5 |
|---|---|---|---|---|---|---|---|
| f(x) |
Sketch the graph of f \left( x \right) = e^{x} for -5\leq x\leq 5.
Consider the function f\left(x\right) = e^{ - x }.
Can the function ever have a negative value?
As the value of x gets larger and larger, what value does f\left(x\right) approach?
As the value of x gets smaller and smaller, what value does f\left(x\right) approach?
Can the value of f\left(x\right) ever be equal to 0?
Find the value of f\left(x\right) when x=0.
How many x-intercepts does the function have?
Sketch the graph of f\left(x\right) = e^{ - x }.
Sketch both y = e^{x} and y = e^{ - x } on the same set of axes. What are the coordinates of their intersection point?
Consider the function f(x) = - e^{x}.
Complete the following table of values. Round each value to three decimal places.
| x | -9 | -6 | -3 | 0 | 3 | 6 | 9 |
|---|---|---|---|---|---|---|---|
| f(x) |
Are there any values of x where f\left(x\right) is positive?
Are there any values of x where f\left(x\right) is equal to 0?
Is the function increasing or decreasing?
How would you describe the rate of increase or decrease of the function?
Using a graphing calculator, plot each set of three graphs below on the same screen and determine whether or not they share the same:
y-intercept
Asymptote
Range
For each of the following functions, state whether they are increasing or decreasing:
f_1(x) = e^{x}
f_2(x) = e^{-x}
f_3(x) = -e^{x}
f_4(x) = -e^{-x}
Describe a sequence of transformations to apply to the graph of f\left(x\right) = e^{x} to achieve the graphs of the following:
Find the equation corresponding to the graph of y = e^{x} after it has undergone the following transformations:
Translated three units upward and four units to the left.
Translated five units downward and two units to the left.
Dilated by a factor of two vertically and then translated five units downward.
Reflected across the x-axis, translated three units upward and then two units to the right.
Dilated by a factor of seven horizontally and then translated two units downward.
Dilated by a factor of one third horizontally and then translated two units downward.
For each of the following transformations on y=e^x find:
The equation of the resulting function.
The equation of the resulting horizontal asymptote.
The value of the resulting y-intercept.
The graph of the resulting function.
y is first dilated by a factor of 3 vertically and then translated 2 units upwards.
y is first dilated by a factor of 3 vertically and then translated 1 unit to the right.
y is first translated 2 units upwards and then dilated by a factor of 3.
y is first translated 1 unit to the right and then dilated by a factor of 3 vertically.
y is first reflected across the y-axis and then translated 3 units upwards.
y is first reflected about the x-axis and then translated 2 units to the right.
State the equation of the asymptote of:
Sketch f\left(x\right) = e^x, g\left(x\right) = 2e^x + 1, and h\left(x\right) = e^{2x}-1 on the same set of axes.
Sketch the curves y = e^{x}, y = e^{x} + 3, and y = e^{x} - 4 on the same number plane.
Sketch the graph of y = 5 - e^{2 - 3x}.