EXAMPLE 1 Find the linearization of the function f(x) = Vx + 7 at a = 2 and use it to approximate the numbers 8.95 and 9.05. Are these approximations overestimates or underestimates? SOLUTION The derivative of f(x) = (x + 7)/2 is f'(x) = and so we have (2) = O and f'(2) - . Putting these values into this equation, we see that the linearization is L(x) = f(2) + l'(2)(x – 2) - + (x - 2) -7 Video Example) The corresponding linear approximation is Vx+7 = (when x is near 2). In particular, we have V8.95 =+ (round to four decimal places) and V9.05 = (round to four decimal places). The linear approximation is illustrated in the figure to the left. We see that, indeed, the tangent line approximation is a good approximation to the given function when x is near 2. We also see that our approximations are overestimates because the tangent line lies above the curve. Of course, a calculator could give us approximations for v8.95 and v9.05, but the linear approximation gives an approximation over an entire interval.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
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