Question

If the tangent line to y = f(x) at (6, 3) passes through the point (0, 2), find f(6) and f '(6). f(6) = Incorrect: Your answer is incorrect. f '(6) = Correct: Your answer is correct.

Answer 1

f(6) = 3

f'(6) = 1/6

Explanation:

Remember that for a function f(x), we define f'(x) as the slope of the tangent line to the point (x, f(x))

We know that:

y = f(x) passes through the point (6, 3)

Then we already know that:

f(6) = 3.

Now we also know that the tangent at this point, also passes through (0, 2)

Remember that a line can be written as:

y = a*x + b

Where in this case, a = f'(6)

so we just want to find the slope of this line.

Remember that for a line that passes through (x₁, y₁) and (x₂, y₂) the slope is given by:

a = (y₂ - y₁)/(x₂ - x₁)

And we know that the tangent line passes through the points (0, 2) and (6, 3)

Then the slope is:

a = (3 - 2)/(6 - 0) = 1/6

Then we have:

a = f'(6) =1/6