Estimate f(5.9) given that f(6)=5 and f′(6)=3.
df/dx = 3, so df = 3 dx
dx = -0.1, so ...
-0.3?
I did a question before and it was
Estimate f(2.9) given that f(3)=5 and f′(3)=4.
And the answer was 4.6
df = 4 dx
so if you go from 3 to 2.9 that is
dx = -.1
so
df = -.4
and 5 - .4 = 4.6 sure enough
To estimate f(5.9), we can use linear approximation. The linear approximation formula is:
L(x) = f(a) + f'(a) * (x - a)
In this case, a = 6, f(6) = 5, and f'(6) = 3. We want to estimate f(5.9), so x = 5.9.
L(5.9) = f(6) + f'(6) * (5.9 - 6)
L(5.9) = 5 + 3 * (5.9 - 6)
Now, let's compute the value:
L(5.9) = 5 + 3 * (-0.1)
L(5.9) = 5 - 0.3
L(5.9) = 4.7
Therefore, the estimated value of f(5.9) is approximately 4.7.