Given that f(-0.5)=2 and f'(-0.5)=4, using tangent line approximation you would estimate f(0) to be:
A.4
B.-3
C.0
D.1
E.-2
We can use the tangent line approximation formula:
f(x) ≈ f(a) + f'(a)(x-a)
where a is the point we know information about, which in this case is -0.5, and x is the point we want to estimate, which is 0.
Plugging in the given values, we get:
f(0) ≈ f(-0.5) + f'(-0.5)(0-(-0.5))
f(0) ≈ 2 + 4(0.5)
f(0) ≈ 4
Therefore, the answer is A.4.
To estimate the value of f(0), we can use the tangent line approximation or the linear approximation formula.
The linear approximation formula is given by:
f(x) ≈ f(a) + f'(a)(x - a)
In this case, we are given that f(-0.5) = 2 and f'(-0.5) = 4. We want to estimate f(0), so a = -0.5 and x = 0.
Using the linear approximation formula, we can plug in the values:
f(0) ≈ f(-0.5) + f'(-0.5)(0 - (-0.5))
f(0) ≈ 2 + 4(0.5)
f(0) ≈ 2 + 2
f(0) ≈ 4
Therefore, the estimate for f(0) using the tangent line approximation is 4.
The correct answer is (A) 4.