Answer:
To estimate the relative rate of change of f(t) = 4t at t = 5 using At = 0.01, we need to calculate the difference quotient:
f'(5) ≈ (f(5 + At) - f(5))/(At)
f'(5) ≈ (f(5.01) - f(5))/(0.01)
Substitute f(t) = 4t into the equation:
f'(5) ≈ (4(5.01) - 4(5))/(0.01)
f'(5) ≈ (20.04 - 20)/(0.01)
f'(5) ≈ 0.04/0.01
f'(5) ≈ 4
Therefore, the relative rate of change of f(t) = 4t at t = 5 is approximately 4.
To estimate f(26), we can use f'(25) ≈ (f(26) - f(25))/(1):
-0.5 ≈ (f(26) - 3.4)/(1)
Solving for f(26):
f(26) ≈ -0.5 + 3.4
f(26) ≈ 2.9
Therefore, f(26) is approximately 2.9.
To estimate f(30), we can use f'(25) ≈ (f(30) - f(25))/(5):
-0.5 ≈ (f(30) - 3.4)/(5)
Solving for f(30):
f(30) ≈ (-0.5)(5) + 3.4
f(30) ≈ -2.5 + 3.4
f(30) ≈ 0.9
Therefore, f(30) is approximately 0.9.