7. Applications: Derivatives of Logarithmic and Exponential Functions

by M. Bourne

We can now use derivatives of logarithmic and exponential functions to solve various types of problems eg. in the fields of earthquake measurement, electronics, air resistance on moving objects etc.

Cessna taking off

Cessna taking off. [Credit: Photobunny]

Example

A Cessna plane takes off from an airport at sea level and its altitude (in feet) at time t (in minutes) is given by

h = 2000 ln (t + 1).

Find the rate of climb at time t = 3 min.

Note: In aviation, height above sea level is meaured in feet. It is regarded as a metric unit and is used universally in aviation instrumentation and charts. Problems can occur when civilian charts show heights of mountains in metres.

Sound Pressure and Decibels

The sound pressure P for a given sound is given by:

`P=10\ log\ W/(W_0`

Its units are decibels (dB).

W is the size of a variable energy source (called the sound power), measured in Watts.

Wo is the lowest threshold of sound that humans can typically hear. It is a constant given by:

`W_0=10^(-12)\ "W/m"^2`.

The sound pressure is related to the sound intensity of a sound wave. Logarithms are used to cope with the large variation in sound pressure that humans can hear (from the whispering wind at around 20 dB up to the roar of a rock concert at 120 dB, depending on the distance from the speakers).

Exercise 1a

Find the rate of change of the sound pressure `P` with repect to time if `W = 7.2` and `(dW)/dt = 0.5` at some given time `t`.

Exercise 1b

If the variable sound power W is given by

W = t2 + t + 1,

find the rate of change of the sound pressure P, at time t = 3 s.

Exercise 1c

If W = cos 0.2t, find the rate of change of the sound pressure P, at time t = 1 s.

Exercise 2

The charge of a capacitor in a circuit containing a capacitor of capacitance C, a resistance R, and a source of voltage E is given by

`q=CE(1-e^({:-t:}//{:RC:}))`

Show that this equation satisfies the equation

`R(dq)/(dt)+q/C=E`

RC circuits in electronics are an important application of differentiation (and differential equations, which you meet later). In the question we just completed, we showed that the expression for charge satisfies a particular differential equation. If you are brave, you can have a sneak preview of how this all works in Application of Ordinary Differential Equations: RC Circuits.

Exercise 3

The radius of curvature at a point on a curve is given by

`R=([1+((dy)/(dx))^2]^(3/2))/((d^2y)/(dx^2))`

A roller mechanism moves along a path defined by:

`y = ln(sec\ x)` for `-1.5\ "dm" < x < 1.5\ "dm"`.

Find the radius of curvature of this path for `x = 0.85\ "dm"`.

[A "dm" is a decimeter, or 1/10 of a meter.]

See other examples of radius of curvature.

Exercise 4

A computer is programmed to inscribe a series of rectangles in the first quadrant under the curve of

`y=e^-x`

What is the greatest area of the largest rectangle that can be inscribed?

[For a reminder on graphing exponential functions, see Graphs of Exponential and Logarithmic Functions.]

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