Note: Answers should be in decimal form. Up to two decimal places only.

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146=144+2=144(1+2/144)=144(1+1/72).

√146=√(144(1+1/72))=12(1+1/72)^½.

This approximates to 12(1+(1/2)(1/72) linearly= 12(1+1/144)=12+1/12=12.08 to 2 dec places.

To use linear approximation formally let y=√(144+x), dy/dx=1/(2√(144+x)). If x=a=0, y=12, and dy/dx=1/24.

L(x)=12+x/24. When x=2, L(2)=12+1/12=12.08 as before.

Alternatively:

f(x)=√x, df/dx=1/(2√x); a=144, so df/dx=1/24 at x=a.

f(x)=12; L(x)=12+(1/24)(x-144)=12+x/24-6=6+x/24.

f(146)≃L(146)=6+146/24=(144+146)/24=290/24=145/12=12+1/12=12.08 as before.

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