Applications of derivatives 03/25/16
Derivatives have a myriad of applications in all of science. One of the most practical uses of differential calculus is the evaluation of newtonian mechanics. The most fundamental of facest, kinematics, is simply derivatives. Velocity is the derivative of position in respect to time, or v(t)=dxdt, and acceleration is the derivative of velocity in respect to time, or a(t)=dvdt=d2xdt2. This means that if our function is a defined variable, then we can find all sorts of information about it by applying the fundamental rules of calculus
Another interesting application of differential calculus is the mean value theorem. The mean value theorem states that if we have a function whose graph is continuous on [a,b]and differentiable on (a,b), and that f(a)=f(b), then f(x)’must be equal to 0 at some point in that line. This makes perfect intuitive sense because if a graph goes up then it must come down or vice versa, and if is continuous and differentiable then the sign of the derivative must change and when it changes it hits 0.
A further application includes Linear approximation. If we have a very complicate function to evaluate, then we can take the derivative at a known point and evaluate it further from there. Because the value of derivatives do not change much for close values, we can use a linear tangent from a known variable to make a fairly accurate approximation. The formula for this equation is y=y0 +f(x0)*x.Take for example the function f(x)=x. It is rather difficult to find the numerical value for 1.1off hand, however, it is extremely easy to find f(1)’ as well as f(1). from there, we can evaluate it using our equation which becomes 1+12*1-1/2*0.1=1.05This is extremely close to the true value of 1.0488. In fact, there is only a 0.114%difference! Truly remarkable!
Another useful feature of Differentiation can be used in finding critical points. A critical point is where the derivative of a function f(x)’=0 or f(x)’=undefined When this happens, a graph of a function will either start to decrease or increase. We can find these point using calculus, and then we can determine if it is a maxima or a minima by proceeding with the following steps. If the f(x)” is positive, then it is a minima because the function was decreasing until the point c where it changes signs, and vice versa for a minima