## Calculus Without Tears## Synopsis of Volume 3 - Nature's Favorite Functions## Differentiating PolynomialsPolynomials are engineers' favorite functions because they are easy to calculate. First we complete our study of the calculus of polynomials. (Note: polynomials are not nature's favorite functions however, see the Second Order Systems section below.) If Here's why: think of Thus, This is the product rule for derivatives and can be used to easily determine the derivative for any power of t, and hence for any polynomial. ## Polynomial Approximation and Taylor's TheoremPolynomials are easy to calculate. Functions other than polynomials, like radicals, exponentials, and trigonometric functions, are generally impossible to calculate directly. The solution to this delimma: approximate these functions with polynomials. Taylor's theorem makes this possible. The linear approximation to a function matches the value of the function and the 1st derivative of the function. A linear function is a degree 1 polynomial. The n (Check that Taylor's theorem gives a bound on the error of the approximation as t moves away from 0. We show how to construct polynomial approximations to exponential and trig functions below. Quick proof of Taylor's theorem: the n Here's the country version of this proof: if your old hound's maximum acceleration is M, and he runs for t seconds, than at most he has reached a velocity of M*t, n'est pas? So, he has travelled at most (M*t)*t. Had it been the hound's maximum 'jerk' (rate of change of acceleration) that was M, then, by the same argument, the hound's max acceleration at time t would be M*t, the hound's max velocity M*t*t, and the hound's max distance M*t*t*t. This is the basic principle underlying Taylor's theorem. (It only takes a little finagling to get the denominator (n+1)! as shown above ). ## The Fundamental Theorem of CalculusThe Fundamental Theorem of Calculus (FTOC) is, succinctly,
If And, believe it or not, all the work has been done. We can approximate any velocity function Let ## Good News, Bad NewsThe good news is that we have now covered the calculus of polynomials. The bad news is that the functions that describe nature are usually not polynomials. Let's look back for a minute - in Vol. 1 we analyzed the motion of a runner running at constant velocity, this example is very 'unnatural' as a real runner accellerates and decelerates throughout a race. In Vol. 2 we analyzed the trajectory of a falling object, but to get the result we wanted, a polynomial solution, we had to assume that gravity was constant, and this is another unnatural assumption, as gravity is not constant. In the last chapter of Vol. 2 we studied simple circuits, and I could not find a circuit that produced a polynomial solution for current or voltage, so we could not solve even the simplest circuit analytically, and had to use numerical methods (note: this is what usually happens in engineering). So, if polynomials do not occur in nature, what functions do? Stay tuned. ## Roots and RadicalsIt's easy to define the meaning of fractional exponents. For example, y ## Exponentials and LogarithmsIf the exponent is the variable, we have an exponential function, e.g. The natural log function is the inverse of exp, that is ln(exp(t)) = t. We use the chain rule to show that the derivative of ln(t) is 1/t. We know the derivatives of all orders for 1/t, so we can construct polynomial approximations to ln. The values of exp and ln are calculated by using polynomial approximation. Ln and exp can then be used to calculate the values of fractional exponents.
## Trigonometric FunctionsThe basics of plane trigonometry are covered in five lessons, with the goal of defining the sine and cosine functions, and deriving the trigonometric identity needed to differentiate them ( Using this identity the derivative of Values of trig functions are calculated using polynomial approximation. We have ## Second Order SystemsLinear second order systems are used to model a wide variety of physical phenomena, and are the basic building blocks of engineering analysis. The differential equation for a linear second order system can be written (t) + k*p'(t) = pf(t), with all the terms containing a derivative of p on the left side and the 'forcing function' f on the right. If the forcing function is replaced by 0 we have the homogeneous form of the equation; this equation characterizes the unforced motion of the system. It is 'homogeneous' because all the terms contain p or a derivative of p. Exponential and trigonometric functions provide the solutions to homogeneous DEs, hence, they are nature's favorites, and they figure in almost every engineering analysis.Analyzing the spring mass assembly shown in the figure, we start with F=MA and let M=1 for simplicity.
The forces acting on the block are gravity and the force of the table pushing up on the block, which cancel, and the forces due to viscous damping (-c* (t)) where u is the damping coefficient and K is the spring constant. So, the differential equation for the unforced system is
p(t) + c*p''(t) + k*p'(t) = 0pIn the case where the spring is strong, i.e., k In the case where the spring isn't so strong, i.e., k To analyze the circuit in the diagram we use Kirchoff's voltage law, which states that the sum of
the voltages around a closed loop is 0. The equation characterizing the inductor is i'(t) where v is the voltage across the inuction, L is the inductance of the inductor, and _{L}i is the current in the loop. The equation characterizing the resistor is v(t) = R*_{R}i(t) where v is the voltage across the resistor and R is the resistance of the resistor. The equation characterizing the capacitor _{R}v'(t) = 1/C _{C}i(t) where v is the voltage across the capacitor C is the capacitance of the capacitor. Kirchoff's voltage law is _{C}v + _{L}v + _{R}v = 0, that is_{C}L* i'(t) + R*i(t) + = 0differentiating both sides and dividing by L yields (t) + (R/L)*i''(t) + (1/LC)*i'(t) = 0iWait a minute, this differential equation is mathematically identical to the one for the spring-mass assembly, the only things having changed are the names of the function, variables and constants. So, the solutions are the same. |