Suppose we have a function f(x) such that f'(1) = 3. Then the value of the following limit: lim→0 (f(1+2h)-f(1-h))/7h?
回答 (2)
I assume you meant lim h→0 [f(1+2h)-f(1-h)]/(7h)
[f(1+2h)-f(1-h)]/(3h) would be the slope of the secant line through the points (1-h, f(1-h)) and (1+2h, f(1+2h))
This slope will approach 3 as h approaches 0 since f'(1) = 3.
But the limit you are asked to find has 7h in the denominator instead of 3h which means your limit will be 3/7 as much.
lim h→0 [f(1+2h)-f(1-h)]/(7h) = 9/7
I guess I'd use the linear approximation for f(x) near 1, which is that you take the value at x = 1, f(1). If you are shifting by an amount h, then y changes by f'(1) * h.
That is, f(1 + h) = f(1) + h * f'(1) (approximately, and the difference between those two expressions goes to 0 as h->0).
With that in mind, replace f(1 + 2h) by f(1) + 2h * f'(1).
Replace f(1 - h) by f(1) - h * f'(1).
Simplify.
收錄日期: 2021-04-24 01:13:11
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