Final answer:
Estimating f'(1) using a positive difference quotient for the function f(x) = 4log(x) involves using the definition of a derivative and substituting a value close to 1 into the formula, leading to an approximate result of 0.0172.
Step-by-step explanation:
To estimate f'(1) for the function f(x) = 4log(x) using a positive difference quotient, we'll consider a value of x that is slightly larger than 1. Let's choose x = 1 + h, where h is a small positive number. By the definition of a derivative at a point, we have:
f'(1) ≈ (f(1+h) - f(1)) / h
Substitute f(x) into the equation:
f'(1) ≈ (4log(1+h) - 4log(1)) / h
Since log(1) = 0, the equation simplifies to:
f'(1) ≈ 4log(1+h) / h
To estimate this further, we could let h = 0.01 for a good approximation. Thus:
f'(1) ≈ 4log(1.01) / 0.01 = approximately 4(0.0043) = 0.0172