Linear Approximation of a Function at a Point. Consider a function \(f\) that is differentiable at a point \(x=a\). Recall that the tangent line to the graph of \(f\) at \(a\) is given by the equation …

Consider the second derivative as a difference quotient in terms of the first derivative: f′′(x) = f (x+h)−f (x) h. Substitute f′(x + h) = f(x+2h)−f(x+h) h and f′(x) = f(x+h)−f(x) h to acquire: f′′(x) = …

Example 1 Linear Approximation of the Square Root . Let $f(x) = x^{1/2}.$ Find the linear approximation of $f$ near $x = 4$ (at the point $(4, f(4)) = (4, 2)$ on the graph), and use it to …

A simple approximation of the first derivative is f0(x) ≈ f(x+h)−f(x) h, (5.1) where we assume that h > 0. What do we mean when we say that the expression on the right-hand-side of (5.1) is an …

Estimate p 26 using a linear approximation. Solution. Let f(x) = x. Then we wish to approximation p f(26). To do this, look for a nearby input to this function that we can compute easily: f(25) = …

To get the quadratic term, we just need to make sure that the rst and second derivative at x = a agree. This gives the formula Q(x) = f(a) + f0(a)(x a) + f00(a)(x a)2=2. Indeed, you can check …

example: Find a quick approximation for p 1:1 without a calculator. Clearly, this is close to p 1 = 1, but we want more accuracy. Take f(x) = p x, so f0(x) = 1 2 x 1=2 and f0(1) = 1 2. For xnear a= …

In the next example we will nd another local linear approximation. Then we’ll inves-tigate analytically how accurate the line is in estimating the curve. We’ll need the second derivative …

Example: Use the apparatus of linear approximation to provide an approximate value of 28 . Solution: Consider the differentiable function fx x() = . Its derivative is 1 2 fx x ′ = and both of …

We have seen how to solve a restricted collection of differential equations, or more accurately, how to attempt to solve them---we may not be able to find the required anti-derivatives. Not …