If the tangent line to y = f(x) at (4,3) passes through the point (0,2), Find f(4) and f'(4)? An explanation would also be very helpful.

Solution:

Given that the tangent line to y = f(x) is at (4, 3)

The point lies on the curve f(x).

⇒ The point must satisfy the curve y = f(x).

⇒ f(4) = 3

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.

Also given that the tangent at (4, 3) passes through the point (0, 2).

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 (4, 3) and (0, 2) with x\(_1\) = 4, y\(_1\) = 3 and x\(_2\)= 0, y\(_2\) = 2 is (2 - 3)/(0 - 4) = -1/-4 = 1/4

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

Therefore, f'(4) = 1/4.

If the tangent line to y = f(x) at (4,3) passes through the point (0,2), Find f(4) and f'(4)? An explanation would also be very helpful.

Summary:

Navigation on the ocean is an important use of tangents. If the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), then f(4) = 3 and f' (4) = 1/ 4 .