suppose that f(x) is a function of f(125)=52 and f'(125)=4. estimate f(123.5)
f(x+dx) = f(x) + f'(x) dx
f(125) = 52
f'(125) = 4
f(125-1.5)= 52 + 4(-1.5)
Well, if I were to estimate f(123.5) based on the given information, I would say it's probably somewhere between "don't worry about it" and "ask a fortune teller." But in all seriousness, we can use the information we have to make an estimate. Since f'(125) tells us the rate at which f(x) is changing at x = 125, we can assume that it changes at a relatively constant rate in that small interval. So, we could approximate f(123.5) by adding 4*(123.5-125) to f(125). That would give us f(123.5) ≈ 52 + 4*(-1.5) = 52 - 6 = 46.
Now, remember, this is just an estimation based on the given information, so take it with a grain of clown dust!
To estimate the value of f(123.5), we can use the concept of linear approximation. Linear approximation relies on the approximation that close to a point x=a, the function f(x) can be approximated by a linear function, which is given by:
L(x) = f(a) + f'(a)*(x-a)
In this case, we want to estimate f(123.5), so we can use a = 125 as the point of approximation. Plugging the values into the linear approximation formula, we have:
L(123.5) = f(125) + f'(125)*(123.5 - 125)
L(123.5) = 52 + 4*(-1.5)
L(123.5) = 52 - 6
L(123.5) = 46
Therefore, based on the linear approximation, we estimate that f(123.5) is approximately equal to 46.
To estimate the value of f(123.5), we can use a technique called linear approximation or the tangent line approximation. This involves using the information about the function and its derivative at a nearby point to approximate the value at the desired point.
Here's how you can estimate f(123.5) using linear approximation:
Step 1: Find the equation of the tangent line to the graph of f(x) at x = 125.
Since f'(x) represents the slope of the tangent line at any point x, we can use the point-slope form of a linear equation to find the equation of the tangent line.
The slope (m) of the tangent line is given by f'(125), which is 4. So we have:
m = 4
We also have a point on the tangent line (125, f(125)) = (125, 52). Plugging these values into the point-slope form, we get:
y - f(125) = m(x - 125)
y - 52 = 4(x - 125)
y - 52 = 4x - 500
y = 4x - 448
Therefore, the equation of the tangent line is y = 4x - 448.
Step 2: Use the equation of the tangent line to estimate f(123.5).
To estimate f(123.5), we substitute x = 123.5 into the equation of the tangent line:
y = 4x - 448
f(123.5) = 4(123.5) - 448
f(123.5) = 494 - 448
f(123.5) = 46
Therefore, the estimate for f(123.5) is 46.
Note: Keep in mind that this is an estimate based on the assumption that the function behaves linearly in the vicinity of the point (125, 52). It is not an exact value, but rather an approximation.