What is linear approximation used for? Why would you use the linear approximatio

Question

What is linear approximation used for? Why would you use the linear approximation of a function rather than the actual function? State the general formula for La(x), the linear approximation of a function around the point x = a. Use this formula to find the linear approximation L0(x) of the function f(x) =ROOT 1 - x at a = 0. Use the linear approximation that you found in to approximate ROOT 0.99. Use your calculator to find the actual value of ROOT 0.99 correct to 5 decimal places. On the axes below sketch and label the graphs of both f(x) and L0(x) and label the point where they touch. Use these graphs to determine if your estimate in part (c) is greater or less than the actual value of ROOT 0.99. Explain your choice by referring to the graphs of f(x) and L0(x). Indicate your answer on the graphs you have drawn. An incorrect use of L'Hospital's Rule is illustrated in the following limit computation. Explain what is wrong and find the correct value of the limit. Lim(x pi/2) sin x/x = lim (x pi/2) cosx/1 = 0 Find lim (x -infinity) xe x using L'Hospital's Rule. Using this information, does the function f(x) =x e x have a horizontal or vertical asymptote? State the equation of this asymptote.

Explanation / Answer

Ok. Linear approximations are used to find approximate values for a system output when the independent variable is only changing by a very small amount. Imagine taking the derivative of some function at a single point. You will get a tangent line -- if you move along the tangent line just a little bit, you will notice that the linear values for "lets say x" would produce close approximations for y. This is the purpose for taking derivatives, learning the Taylor, McLaurin and Fourier series, Ordinary Differential Equations, Laplace Transformations, etc...

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