Linear Approximations
Engineering Context
Engineers use mathematics as a toolbox to analyze the world around them and design a better tomorrow. Linear approximation is defined as a result that is not exact, but is still close enough to be used. This is perfect for engineering. Linear approximations allow us to analyze complicated functions and predict an outcome, using simple means. Applications of this concept are limitless, as everyday phenomenons are rarely represented by simple, linear functions, and more often resemble curved polynomial and wave functions. For example, light, sound, and vibrations are represented by wave functions, which can be decomposed and analyzed by linear approximation.
MAE: Pendulums and Angle of Attack
(1) The formula for the period of swing of a simple gravity pendulum is given as follows:
Where θo is the amplitude, or the maximum angle at which the pendulum swings from the vertical. If this amplitude is limited to small swings, \( \theta_o \ll 1 \), we can take the linear approximation of this equation, which gives us:
With such linear approximation, the period is independent of the amplitude of each swing. This is why pendulums are so good at timekeeping.
(2) In aeronautics, the coefficient of lift is determined by the angle of attack, or pitch angle. This is represented by a curved function:
Figure 1
When pitch angles are small, the graph resembles a straight line and engineers analyze it using linear approximation. Therefore, the equation for evaluating coefficient of lift versus angle of attack is:
Where \( C_{L_\alpha} = \frac{dC_L}{d\alpha} \)
ECE: Electrical Resistivity
(1) Electrical resistivity, ρ, measures how strongly a material resists electric current flow. Such resistivity in most materials changes with temperature; as temperature increases, ρ decreases. If temperature does not vary too much, electrical resistivity is analyzed using linear approximation:
Where
\( \alpha = \) temperature coefficient of resistivity
\( T_o = \) fixed reference temperature (usually room temperature)
\( \rho_0 = \) resistivity at temp \( T_o \)
BE: Geometric Optics
Geometric optics is a technique that exists in many medical devices, including medical imaging and surgical machines. When working with geometric optics, engineers use the assumption that for small angles θ, \( \sin(\theta) = \theta \). This assumption can only be made using linear approximation, as follows:
(1) Let \( f(x) = \sin(x) \) . Because we are considering small angles, our linear approximation will be about a = 0.
(2) We know that \( f'(x) = \cos(x) \), so we have:
Thus, using linear approximation, we can conclude that near \( \theta = 0\), sin θ ≈ θ.
Figure 2: \( \sin(x) \) and x
CEE: Population Growth
(1) The population growth formula is used in civil and environmental engineering to predict future needs in an area.
(2) We can use linear approximation as an alternative to evaluate the population growth formula and predict the population at a given time.
The Essentials
A linear approximation is the predicted output of a curved function at given value x using a tangent line, as represented in Figure 3.
Figure 3
The linear approximation is given as:
General Form:
Where
\( f(a) = \) function of a curve
\( f'(a) = \) first derivative of curve function or tangent line
\( (a) = \) chosen center-point
The slope of a curve on a very small scale is close to that of a straight line. We are given a general function f(x) and asked to find the output at a given x value. Because the graph is curved, we can estimate the output at this x value by creating a tangential line.
Step One Find the slope of the tangential line by deriving the given function f(x).
Step Two Establish the a value as the point at which both lines touch. This value should be close in value to x
Step Three Solve for \( f(a) \) by plugging in the chosen a value to the original function.
Step Four Solve for \( f'(a) \)
Step Five Plug each piece into the formula and solve for y. The resultant value is an approximation of the original function, using a tangent line.
A Deeper Dive
When dealing with linear approximations, engineers are hoping to predict how a function behaves at a specific point using a tangent line. Let’s take a look at the symbolic description of the linear approximation of a function, given as:
If we rearrange this equation just slightly, we can easily compare it to the basic formula for a straight line.
Thus, \( f'(a) \) correlates to m, or the slope of a straight line. This makes sense because a derivative \( f'(a) \) is the instantaneous rate of change of a function with respect to a variable at a specific point, otherwise known as the slope. Thus, \( f'(a) \) correlates with m, the slope of a straight line. And \( f(a) \) will produce a constant, correlating with b, which is the y-intercept of said straight line.
How can a straight line be an accurate approximation of a curved line? After all, a straight line doesn’t curve. To answer this question, it is important to know that linear approximations are only accurate for x values that are very close to the chosen center-point a. Let’s visualize this concept by looking at the function \(f(x) = \sqrt{x}\) with its linear approximation, centered at point (1, 1).
Figure 4
At a wide domain of x values, the linear approximation does not appear to evaluate the curve very well. But let’s see what happens when we zoom in closer to the center-point (1, 1).
(a)
(b)
As seen in figure b, linear approximations are accurate at x values very close center-point a. The greater the difference between x and a, the less accurate the approximation will be.
Interestingly, the idea of linear approximations is expanded in Calculus II when dealing with Taylor Series. A Taylor Series is a method of summation used to approximate the curve of a function, based on a given center point. The formula is:
Do the first two terms look familiar? The beginning of a Taylor series is in fact a linear approximation! In other words, a linear approximation is a truncated version of a Taylor series, evaluated only at n = 0 and n = 1. This is because as n approaches infinity, the changes in each term get smaller and smaller, meaning that evaluations beyond n = 5 are generally not very effective. The term for n = 2 would introduce curvature to our estimation graph, but for the purposes of generating a straight line, n = 0 and n = 1 are sufficient.
Practice
Exercise 1. Find a linear approximation to \( h(x) = x^4 - 6x^3 + 3x - 7 \), at \(x = -3\). Then use to predict the value at \(x = 3.1\)
Figure 6
Exercise 2. Find a linear approximation to \( f(x) = 3x e(2x - 10) \) at \(x = 5\).
Solution 1. We know that the formula for linear approximation is:
Using the step-by-step from above, we will first find h′(x):
We know that:
Plug in a to find h(a).
Plug in a to find h′(a).
Now, we have all of the pieces necessary to set up our linear approximation:
Distribute and simplify to get:
We are asked to approximate at \(x = −3.1\), so we will plug this in and solve.
Thus,
Solution 2.