If the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), find f(4) and f'(4).
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To find f(4) and f'(4), we can use the information provided about the tangent line and the point it passes through. Here's how:
1. Let's start by finding the slope of the tangent line. We know that the tangent line passes through the points (4, 3) and (0, 2). The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
Substituting the given points:
slope = (2 - 3) / (0 - 4)
= (-1) / (-4)
= 1/4
Therefore, the slope of the tangent line is 1/4.
2. Since the tangent line is tangent to the curve at (4, 3), its slope is the same as the derivative of the function f(x) at x = 4. So, we can write:
f'(4) = 1/4
Therefore, the derivative of f(x) at x = 4 is 1/4.
3. Now, let's find the equation of the tangent line using the point-slope form. We know the slope of the line is 1/4, and it passes through the point (4, 3). The point-slope form is given by:
y - y1 = m(x - x1)
Substituting the values:
y - 3 = (1/4)(x - 4)
Simplifying the equation:
y - 3 = (1/4)x - 1
4. To find f(4), we need to substitute x = 4 into the equation y = f(x) and solve for y. Since the tangent line passes through (4, 3), f(4) = 3. Therefore:
f(4) = 3
So, the value of f(4) is 3.
To summarize:
- f'(4) = 1/4
- f(4) = 3