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 technology, 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 across 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}.