When h is sufficiently small:
f(a + h) ≒ f(a) + f'(a)·h（first-order approximation）
This is equivalent to approximating f by its tangent line。
Example ① Calculate √4.04 using an approximation formula.
Solution:f(x)=√x、a=4、h=0.04。f(4)=2、f'(x)=1/(2√x)、f'(4)=1/4。
√4.04 ≒ 2 + (1/4)·0.04 = 2.01
Solution:f(x)=√x、a=4、h=0.04。f(4)=2、f'(x)=1/(2√x)、f'(4)=1/4。
√4.04 ≒ 2 + (1/4)·0.04 = 2.01
Example ② sin 31° ≒ ? (using 30°=π/6 as the reference)
Solution:h = 1°= π/180 ≒ 0.01745。
sin(π/6 + h) ≒ sin(π/6) + cos(π/6)·h = 0.5 + (√3/2)·0.01745 ≒ 0.5151
Solution:h = 1°= π/180 ≒ 0.01745。
sin(π/6 + h) ≒ sin(π/6) + cos(π/6)·h = 0.5 + (√3/2)·0.01745 ≒ 0.5151
Key Point
- Useful when calculating a value that is “slightly different”
- Convert angles to radians
Practice Problems
- Calculate (1.02)¹⁰ using an approximation formula。
Solutions and Explanations
- Solution:1.2
f(x)=x^10、a=1、h=0.02。f'(1)=10。1+10·0.02=1.2。