Antiderivatives and Indefinite Integration Question

Question

Given that the graph of f(x) passes through the point (7,6) and that the slope of its tangent line at (x, f(x)) is 5x+7, what is f(3)?

Answer 1

Hi Jay!

Roger's way is much more concise.

Here is another way.

We're told the tangent line is 5x+7,

so f'(x) = 5x+7

f(x) = ∫f'(x)dx

f(x) = ∫(5x+7)dx

5x²

f(x) = ---- + 7x + C

2

(7,6) is a point on f(x), so can

use this to find C

5*7²

6 = ------ + 7(7) + C

2

245

6 = ----- + 49 + C

2

12 = 245 + 98 + 2C

-331 = 2C

-331

C = ------

2

5x² 331

f(x) = ------ + 7x - ------

2 2

To find f(3), just plug in 3

for x above. You'll find it

should simplify to -122.