antiderivatives question

Question

Given that the graph of f(x) passes through the point (6,7) and that the slope of its tangent line at (x,f(x)) is 3x+2, what is f(2)?

Answer 1

If the slope of the tangent line is 3x + 2 then that means f '(x) or dy/dx = 3x + 2. To find f(x), take the antiderivative:

dy/dx = 3x + 2

∫dy/dx = ∫3x + 2

∫dy = ∫3x + 2 dx

y = 3/2x² + 2x + C

Since we are told "f(x) passes through the point (6, 7)" then we can plug in x = 6 and y = 7 to solve for C:

7 = 3/2(6)² + 2(6) + C

7 = 54 + 12 + C

C = -59

So f(x) = 3/2x² + 2x - 59

To find f(2), plug in x = 2:

3/2(2)² + 2(2) - 59 = -49