# If the tangent line to y = f(x) at (5, 2) passes through the point (0, 1). Find f(5) and f'(5).

**Solution:**

Given that tangent line to y = f( x ) is at (5, 2)

The tangent line is the straight line that passes through a point on the curve and at that point, the tangent line just touches the curve.

⇒The point satisfies the curve equation.

⇒ Point (5, 2) lies on the curve y = f(x) with x = 5 and y = 2

⇒ f(5) = 2

Also given that the tangent at (5, 2) passes through the point (0, 1).

The slope of the line through (x_{1}, y_{1}) and (x_{2}, y_{2}) is (y_{2} - y_{1})/(x_{2} - x_{1})

⇒The slope of the line through (5, 2) and (0,1) with x_{1} = 5, y_{1} = 2 and x_{2} = 0, y_{2} = 1 is (1 - 2)/(0 - 5)

= (-1)/(-5)

= 1/5

Also the slope of the curve f(x) in terms of derivative at a point is f' (x) at that point.

Therefore, f'(5) = 1/5

## If the tangent line to y = f(x) at (5, 2) passes through the point (0, 1). Find f(5) and f'(5).

**Summary:**

If the tangent line to y = f(x) at (5, 2) passes through the point (0, 1) then f(5) = 2 and f'(5) = 1/5. Navigation on the ocean is an important use of tangents.