Question

If f(2) = 3, and f^1(2) = 5, find an equation of the tangent line, and the normal line to the graph of y=f(x) at the point where x = 2

Answer 1

Give me a few minutes

Answer 2

What does "f^1(2) = 5" mean?

Answer 3

It means The first derivative of f at the point (2,5)

Answer 4

Okay, so basically, all we need to do is use y = mx + b. We're given f'(2) = 5 which means the derivative(slope) of f(x) = 5. Therefore, all we have to do is find the equation of the line by inserting the given values:

x = 2
y = 3
m = b

y = mx + b
3 = 5(2) + b
3 = 10 + b
3 - 10 = b
-7 = b

So the equation of the line is y = 5x - 7

Answer 5

Now all we have to do is find the normal (perpendicular) line. If the line we need to find is perpendicular, we know that the slope of the other line will be the negative reciprocal of y = f(x).

So we set it up as y = -1/5(x) + b. Then insert the point (2,3) to find b:
3 = -1/5(2) + b
3 = -2/5 + b
3 + 2/5 = b
15/5 + 2/5 = b
17/5 = b

Equation of the normal line

y = -1/5(x) + 17/5