Given that the graph

Question

Given that the graph of  passes through the point  and that the slope of its tangent line at  is , what is ?

Answer 1

Hello James,

The slope of the tangent line is 7x + 5. that means the derivative of f is f'(x) = 7x+5. From there, you would take the antiderivative of both sides:

f(x) = (7/2)* x² + 5x + C

To find f(2), we first have to find the constant C

We find C by plugging the data from the point (4,7) which the graph of f(x) goes through.

7 = (7/2)*(4²) + 5(4) + C

7 = (7/2)* 16 + 20 + C

7 = 56 + 20 + C

7 = 76 + C

7-76 = C

C = -69

that means: f(x) = (7/2)* x² + 5x - 69

To find f(2), simply replace x by 2 in the previous equation:

f(2) = (7/2) * (2²) + 5(2) - 69

You can take it from here by plugging it in a calculator, make sure you put parentheses as shown, and you will get the right answer.

Cheers.