If the tangent line to y = f(x) at (7, 5)

Question

 passes through the point (0, 4), find f(7) and f '(7)?

Answer 1

The tangent line y = f(x) at (7, 5) passes through the point (0, 4).

Means that, The tangent line y = f(x) passes through the points (7, 5) and (0, 4).

 

Slope - intercept form line equation is y = mx + b, where m is slope and b is y - intercept.

Let the points are (x₁, y₁) = (7, 5) and (x₂, y₂) = (0, 4).

Slope (m) = [(y₂ - y₁)/(x₂ -x₁)]

m = [(4 - 5)/(0 - 7)]

m  = [(-1)/(- 7)]

⇒ m = 1/7.

This is the slope of the tangent line.

Now, the tangent line equation is y = (1/7)x + b.

 

Find the y - intercept by substituting any point in the tangent line equation say (x, y) = (7, 5).

5 = (1/7)(7) + b

b = 5 - 1

⇒ b = 4.

The equation is y = ( 1/7)x + (4).

y = f (7) = ( 1/7)(7) + 4. = 5.

 f (7) = 5.

f(x) = ( 1/7)x + 4

Apply derivative on each side with respect of x.

f '(x) = 1/7

∴ f ' (7) = 1/7.