Question

Let f(x)=lnx.

(a) With the aid of a calculator, use the interval from a=1 to b=1.01(h=0.01) to cstimate f ′ (1).
(b) Improve your accuracy by estimating f ′ (1) with the smaller interval a=1 and b=1.001 (h=0.001).

Answer 1

To estimate f'(1) using the function f(x) = ln(x), we can use the formula for the derivative: f'(1) = lim(h->0) [(f(1+h) - f(1)) / h]. Using different intervals, we can calculate f'(1) with a calculator to improve accuracy.

Step-by-step explanation:

To estimate f'(1) using the function f(x) = ln(x), we can use the formula for the derivative:

f'(1) = lim(h->0) [(f(1+h) - f(1)) / h]

(a) When the interval is from a=1 to b=1.01 (h=0.01), we can substitute these values into the formula and calculate f'(1) using a calculator:

f'(1) = [ln(1.01) - ln(1)] / 0.01

(b) To improve accuracy, we can use a smaller interval from a=1 to b=1.001 (h=0.001) and calculate f'(1) again:

f'(1) = [ln(1.001) - ln(1)] / 0.001