Engineering functions
Engineering models often use a small group of functions again and again. Exponential functions describe
growth, decay, charging, cooling, and damping. Trigonometric functions describe oscillation and rotation.
Logarithmic functions describe scales such as decibels. Hyperbolic functions appear in cable shapes,
beams, and smooth transition models.
Exponential response
A decreasing exponential model has the form
\[
f(t)=Ae^{-kt}.
\]
Here \(A\) is the initial scale and \(k\) controls how quickly the function decays. A larger \(k\) means
faster decay.
A charging or rising response is often written as
\[
f(t)=A\left(1-e^{-t/\tau}\right).
\]
The parameter \(\tau\) is a time constant. After one time constant, the response has completed about
\(63.2\%\) of the way from its initial value to its final value.
Damped oscillation
A common engineering vibration or signal model is
\[
f(t)=Ae^{-kt}\sin(\omega t+\phi).
\]
The sine part creates oscillation. The exponential part reduces the amplitude as time passes.
| Parameter |
Meaning |
| \(A\) |
Initial amplitude scale |
| \(k\) |
Damping or decay rate |
| \(\omega\) |
Angular frequency |
| \(\phi\) |
Phase shift |
Logarithmic engineering scales
Logarithms are used when values cover a very large range. In electronics and acoustics, decibels often use
base-10 logarithms.
\[
G_{\mathrm{dB}}=20\log_{10}\left(\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}\right).
\]
The factor \(20\) is used for voltage, current, pressure, and field-amplitude ratios. For power ratios,
the common form is
\[
G_{\mathrm{dB}}=10\log_{10}\left(\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\right).
\]
Trigonometric functions
Trigonometric functions appear in rotating systems, waves, AC circuits, vibrations, and signal processing.
A simple sinusoidal model is
\[
f(t)=A\sin(\omega t+\phi).
\]
If the ordinary frequency is \(f_0\), then the angular frequency is
\[
\omega=2\pi f_0.
\]
Hyperbolic functions
Hyperbolic functions look similar to trigonometric functions but are based on exponentials. The most common
ones are
\[
\sinh(x)=\frac{e^x-e^{-x}}{2},
\qquad
\cosh(x)=\frac{e^x+e^{-x}}{2},
\qquad
\tanh(x)=\frac{\sinh(x)}{\cosh(x)}.
\]
The catenary shape of a hanging cable can be modeled by
\[
y=a\cosh\left(\frac{x}{a}\right).
\]
Numerical slope
The calculator estimates the local slope with a central-difference formula:
\[
f'(x)\approx\frac{f(x+h)-f(x-h)}{2h}.
\]
This gives a quick estimate of how fast the output is changing near the selected input value.
Parameter studies
A parameter study changes one parameter while keeping the others fixed. This is useful because engineering
design often asks questions like:
- What happens if damping increases?
- How does the response change if the time constant is larger?
- How sensitive is the output to one coefficient?
- Which parameter produces the largest design change?
In the graph, a parameter-family preview helps students see how one parameter changes the shape of the
curve, not only one numerical value.
Common mistakes
- Using degrees instead of radians inside sine, cosine, or tangent.
- Confusing \(\log(x)\) and \(\ln(x)\). In this calculator, \(\log(x)\) means \(\log_{10}(x)\).
- Forgetting parentheses, such as writing
exp(-k*t) instead of exp(-k)*t.
- Evaluating logarithms or square roots outside their real-valued domain.
- Changing a parameter without checking the graph scale.
