18-linear-approximation

# 18-linear-approximation - Math 1a Linear Approximations...

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Math 1a Linear Approximations Fall, 2009 \$ % Linear Approximation A linear approximation is dy dx Δ y Δ x . If y = f ( x ), then this can be re-written as f 0 ( a ) Δ y Δ x = y - f ( a ) x - a or y f ( a ) + f 0 ( a )( x - a ) near the point ( x,y ) = ( a,f ( a )). This is the tangent line: y = f ( a ) + f 0 ( a )( x - a ). Thus the linear approximation is: “near x = a , y is approximately the tangent line to the curve at x = a .” 1 (a) Find the equation of the tangent line to y = ln( x ) at x = 1. (b) Use the equation you found in part (a) to estimate ln(1 . 1). (c) Write down a linear estimate ln(1 + x ) L ( x ) for small values of x . 2 (a) Estimate e - 0 . 2 using the linear approximation near x = 0. (b) Give the full linear approximation for e x near zero. 3 (a) Write down the linear approximation for x near x = 9. (b) Estimate 8 . 94 and 9 . 12. (c) Can you tell whether the estimates you obtained in parts (a) and (b) are too high or too low? Hint:

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18-linear-approximation - Math 1a Linear Approximations...

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