009A Sample Final A, Problem 8

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8. (a) Find the linear approximation to the function at the point .
    (b) Use to estimate the value of .

Foundations:  
Recall that the linear approximation L(x) is the equation of the tangent line to a function at a given point. If we are given the point x0, then we will have the approximation . Note that such an approximation is usually only good "fairly close" to your original point x0.

Solution:

Part (a):  
Note that f '(x) = sec x tan x. Since sin (π/3) = √3/2 and cos (π/3) = 1/2, we have
    
Similarly, f(π/3) = sec (π/3) = 2. Together, this means that
    
                
Part (b):  
This is simply an exercise in plugging in values. We have

    
                        
                        
                        

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