Use linear approximation to approximate f(0.8) given f(x)=2x^3-3x+1
To approximate f(0.8) using linear approximation, we can use the formula:
L(x) = f(a) + f'(a)(x - a)
where a is the known value closest to 0.8 and f'(a) is the derivative of the function evaluated at a.
First, let's find the derivative of f(x), which is f'(x):
f'(x) = 6x^2 - 3
Now, we need to choose a value close to 0.8. In this case, the closest value is 1.
Next, plug in the values into the linear approximation formula:
L(0.8) = f(1) + f'(1)(0.8 - 1)
Calculate the values:
f(1) = 2(1)^3 - 3(1) + 1
= 2 - 3 + 1
= 0
f'(1) = 6(1)^2 - 3
= 6 - 3
= 3
L(0.8) = 0 + 3(0.8 - 1)
= -3(0.2)
= -0.6
Therefore, using linear approximation, f(0.8) is approximately -0.6.