If the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), find f(4) and f'(4).
The slope of the tangent line is, from the coordinates of two points you know on the line,
m = (2-3)/(0-4) = 1/4
For the (straight) tangent line,
y = mx + b = (x/4) + b
3 = (4/4) + b
b = 2
Therefore, the tangent line equation is
y = (x/4) + 2
They have already told you that, for the function f(x), f(4) = 3
f'(4) is the slope of the tangent line at x = 4. We have shown that to be 1/4.
You cannot say what the actual function f(x) is. There are an infinite number of possibilities.
To find f(4), we need to substitute x = 4 into the equation y = f(x).
Therefore, f(4) = f(4).
To find f'(4), we can use the given information that the tangent line to y = f(x) at (4, 3) passes through the point (0, 2).
The slope of the tangent line is equal to the derivative of f(x) evaluated at x = 4.
Using the point-slope form of a line, the slope of the tangent line passing through (4, 3) and (0, 2) is given by:
m = (3 - 2) / (4 - 0)
= 1/4
Therefore, f'(4) = 1/4.
To find f(4) and f'(4), we need to use the information about the tangent line and the given point.
Given that the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), we know that the slope of the tangent line is equal to the slope of the line passing through (0, 2) and (4, 3).
The slope of the line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
Using this formula, we can find the slope of the line passing through (0, 2) and (4, 3):
slope = (3 - 2) / (4 - 0) = 1/4
Since the slope of the tangent line is equal to the derivative of f(x) at x = 4, we can conclude that f'(4) = 1/4.
To find f(4), we can determine the equation of the tangent line at (4, 3) using the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept.
We know the slope is 1/4 and the point (4, 3) lies on the line, so we can substitute these values into the equation and solve for b:
3 = (1/4)(4) + b
3 = 1 + b
b = 3 - 1
b = 2
Therefore, the equation of the tangent line at (4, 3) is y = (1/4)x + 2.
Since the equation of the tangent line represents f(x) at x = 4, we can substitute x = 4 into the equation to find f(4):
f(4) = (1/4)(4) + 2
f(4) = 1 + 2
f(4) = 3
Hence, f(4) = 3 and f'(4) = 1/4.