Page 172 - Calculus Workbook For Dummies

156 Part III: Differentiation

π π
u Estimate sin , that’s one degree of course. The approximation is .
180 180
You know the routine
p x = sinx y - y 1 = m x - x 1i
^ h
_
p 0 = 0 " _ , 0 0i is the po int y - = 1^ x - 0h
0
^ h
p x = cosx y = x
l ^ h
p 0 = 1 " 1 is the slope
l ^ h
π π
Your number is , so you get y = .
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Which shows that for very small angles, the sine of the angle and the angle itself are approxi-
π
mately equal. (The same is true of the tangent of an angle, by the way.) is only ⁄200% too big!
1
180
5
10
v Approximate ln e + 5i. The approximation is 10 + 10 .
_
e
Just imagine all the situations where such an approximation will come in handy!
y - y 1 = m x - x 1i
_
q x = ln xh
^ h
^
1
10 10 10
_
q e i = 10 " _ e , 10i is the po int y - 10 = 10 _ x - e i " the tangent line
e
1 1
q x = x y = 10 _ a e + i e k + 10
10
10
l ^ h
5 -
1 1 e
10
q e i = 10 " 10 is the slope 5
l _
e e = 10 + 10
e
Hold on to your hat. This answer is a mere 0.00000026% too big.