11.12 Defining e and the natural logarithm
Consider the function f \left( x \right) = e^{x}.
Complete the following table of values, correct to two decimal places:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| f(x) |
Sketch the graph of f \left( x \right) = e^{x}.
Sketch the curves y = e^{x}, y = e^{x} + 3, and y = e^{x} - 4 on the same coordinate plane.
Consider the graph of y = e^{x}.
Is the function increasing or decreasing?
Is the gradient to the curve negative at any point on the curve?
What does that tell us about the gradient function?
Describe how the gradient is changing along the curve as x increases.
What does this tell us about how the gradient function, y', changes as x increases?
Hence, what type of function could y' be?
f \left( x \right) = e^{x} and its tangent line at x = 0 are graphed on the coordinate axes:
Determine the gradient to the curve at \\ x = 0.
Evaluate f \left( 0 \right).
What is the relationship between f \left( 0 \right) and f' \left( 0 \right)?
Consider the function f \left( x \right) = e^{x}.
If f \left( 4 \right) = 54.59815, determine f' \left( 4 \right) correct to five decimal places.
If f \left( - 5 \right) = 0.00674, determine f' \left( - 5 \right) correct to five decimal places.
Consider the function y = e^{x}-1.
Point P lies on the curve y = e^{x}-1. If the x-coordinate of P is 4, find the y-coordinate of P.
Determine an expression for the derivative function \dfrac{d y}{d x}.
Find the gradient of the tangent at point P.
What is the largest interval over which the function is increasing?
Consider the curve with equation y = e^{x}.
Determine the value of the gradient m of the tangent to y = e^{x} at the point Q \left( - 1 , \dfrac{1}{e}\right).
Hence find the equation of the tangent to the curve at point Q.
Does this tangent line pass through the point R \left( - 2 , 0\right)?
Find the gradient, m of the tangent to each of the following curves, correct to two decimal places:
The curve y = 7 e^{x} at the point where x = 1.3.
The curve y = - e^{x} at the point \left(1, - e \right).
State the value of x where the gradient of the tangent to the curve y = e^{x} is \dfrac{1}{e^{3}}.
For each of the following points:
Find the gradient m of the tangent to y = e^{x} at this point.
Find a, the angle of inclination of the tangent to y = e^{x} at this point. Express your answers in degrees correct to two decimal places if necessary.
The point where x = 0
The point where x = 5
Use a calculator or other technology to approximate the each of 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}}
Find the value of each of the following correct to four decimal places:
\ln 94
\ln 0.042
\ln 78^{4}
\ln \left( 18 \times 35\right)
Consider x=\ln 31. Find the value of x, correct to two decimal places.
Use the properties of logarithms to express each of the following without any powers or surds:
Use the properties of logarithms to rewrite - 2 \ln x with a positive power.
Rewrite each of the following as the sum and difference of logarithms:
Rewrite each of the following expressions as a single logarithm:
\ln 3 + \ln 5
\ln 24 - \ln 4
\ln 8 - \ln 32
\ln \left( 3 x\right) + \ln \left( 5 y\right)
Which expression is equivalent to 2 \ln \left( 7 x\right) for x > 0?
\ln \left( 14 x\right)
\ln 14 + \ln x
\ln 49 + \ln x
\ln \left( 49 x^{2}\right)
Simplify:
Evaluate each of the following expressions:
\ln e^{3.5}
\ln e^{4}
\sqrt{6} \ln \left(e^{\sqrt{6}}\right)
\ln \left(\dfrac{1}{e^{2}}\right)
The population P of a city increases according to the formula P = 3000 e^{ k t} where t is measured in years and k is a constant.
Find the initial population.
Given the population increases to 8000 in 2 years find the exact value of k.
How many complete years will it take for the population to at least double?
The voltage V in volts across an electrical component decays according to the equation V = A e^{ - \frac{1}{2} t } where A is the initial voltage and t is the time in years. Find the value of t such that V is half of the initial voltage, correct to two decimal places.
The spread of a virus through a city is modelled by the function: N = \dfrac{15\,000}{1 + 100 e^{ - 0.5 t }}, where N is the number of people infected by the virus after t days.
How many people will have been infected after 3 days? Round your answer to the nearest whole number.
How many whole days will it take for at least 4000 people to be infected with the virus?