Use a linear approximation of f(x)=log(x) at x=1 to approximate log(3/4).
Express your answer as an exact fraction; remember that log denotes the natural log.
f'=1/x
so, f(3/4)=f(1)-f(1/4)= -0.125
that's not the right answer. The right answer is a negative half. The answer above is not in fractional form.
To approximate log(3/4) using a linear approximation of f(x) = log(x) at x = 1, we'll start by finding the equation of the tangent line to the curve f(x) = log(x) at x = 1.
The equation of a line can be expressed as y = mx + b, where m is the slope and b is the y-intercept.
To find the slope, we'll need to find the derivative of f(x) = log(x) using the rules of differentiation. In this case, we're dealing with the natural logarithm, so the derivative is:
f'(x) = 1 / x
Evaluating this derivative at x = 1, we get:
f'(1) = 1 / 1 = 1
So the slope of the tangent line at x = 1 is 1.
Now, we'll find the equation of the tangent line by substituting the slope and the point (1, f(1)) into the equation:
y - f(1) = m(x - 1)
Since f(1) = log(1) = 0, the equation becomes:
y - 0 = 1(x - 1)
y = x - 1
Now we can use this equation to approximate log(3/4) by plugging in x = 3/4:
y = (3/4) - 1
y = -1/4
Therefore, log(3/4) is approximately -1/4.