Y x = &lcub &ExponentialE − x 2 erf I x _C1 + &ExponentialE − x 2 _C2 x Įq := &DifferentialD &DifferentialD x y x = piecewise 1 Y x = &lcub &ExponentialE − &ExponentialE x _C1 x Įq := &DifferentialD 2 &DifferentialD x 2 y x + 2 x &DifferentialD &DifferentialD x y x + 2 y x = piecewise 0 Some examples are included in the sections that follow.įirst-order Linear with Piecewise (Nontrivial) CoefficientsĮq := &DifferentialD &DifferentialD x y x + piecewise x The type of equations that one can solve include all first-order methods using integration, Riccati, and higher-order methods including linear, Bernoulli, and Euler. We can solve differential equations with piecewise functions in the coefficients.
Solving Differential Equations with Piecewise H := convert min x 2 − 2, x − 1, ' piecewise ' Thus, it can be converted to a new function g ( x ).
#Evaluating piecewise functions calculator series
However, series can do better when using piecewise. This produces an answer with a superfluous order term. The straightforward calculation of the series of f around x =0 can be computed by using the series command. To determine the highest order of continuity and the problem points, enter:ĭerivatives can be found, and piecewise functions are returned. Isdifferentiable newcubic, x, 3, ' badpoints ' This must be true for splines! However, when we check to see if it is C 3, we obtain For example, in the case of our previous spline function, newcubic, we have We can also determine the differentiability class of a piecewise continuous function. It turns out to be a well-behaved, non-piecewise function. For example,Ĭonvert 1 − x, ' piecewise 'Ĭonvert − signum x 1 − x, ' piecewise ' Other piecewise functions can also be converted to piecewise and be properly manipulated. Newcubic ≔ CurveFitting Spline 0, 1, 2, 3, 0, 1, 4, 3, x Normal convert heavyf, ' piecewise ' Heavyf := x Heaviside 1 + x − x − x 2 Heaviside x − 1 + x 2 Heaviside 1 + x + sin x − 1 Heaviside x − 1 x − 1ĭistributions can be converted back to piecewise functions. Note: This works because discont is able to determine the potential discontinuities of piecewise functions. Where Si(x) is the Sine integral function. Using the same function, f ( x ), find its piecewise derivative.įprime ≔ &DifferentialD &DifferentialD x f Examples of solving DEs will be illustrated later. Such functions can be plotted to determine their behavior.īesides evaluating limits, you can do operations such as computing derivatives, integrating, and solving differential equations with piecewise functions. The next several Maple command lines make use of the following piecewise function:į := piecewise x ≤ − 1, − x, x ≤ 1, x 2, 1 īecause of the division by zero, points such as x = 1 cannot be substituted. Every piece is specified by a Boolean condition followed by an expression. The piecewise function has a straightforward syntax. For now, you should make sure you can read the graphs of functions like those shown in the examples above.This worksheet contains a number of examples of the use of the piecewise function. However, those are studied later on in algebra. Piecewise defined functions and functions with asymptotes often have more going on in the graph. There are some types of functions, where you have to be a little more careful. For example, if we wanted to know the value of \(f\) when \(x = -1\) for the function below, we would just find \(x = -1\) on the x-axis and use the graph to find the corresponding y-value. You can do this for the graph of just about any function.
Using this, we can easily read points off the graph. Remember that in the xy-plane, the x-axis is the horizontal axis, and the y-axis is the vertical axis. Since the notation \(f(2)\) represents the output for the input of 2, we can write this as:
The example says that when \(x = 2\), the output is 10. Since \(x = 2\), we can start by writing: This means that the input of the function is \(x\). When \(x = 2\), the value of a function \(y = f(x)\) is 10. To help you understand this notation, let’s look at a couple of examples. This is read “\(g\) of 2” and represents the output of the function \(g\) for the input value of 2. Another example is something like \(g(2)\). For example, \(f(x)\) is read “\(f\) of \(x\)” and means “the output of the function \(f\) when the input is \(x\)”. Function notation is written using the name of the function and the value you want to find the output for.
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