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 (x₁, y₁) and (x₂, y₂) is (y₂ - y₁)/(x₂ - x₁)

⇒The slope of the line through (5, 2) and (0,1) with x₁ = 5, y₁ = 2 and x₂ = 0, y₂ = 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.