Given the graph, what is f(2)?

Question

Given that the graph of f(x) passes through the point (10,5) and that the slope of its tangent line at (x,f(x)) is 3x+2. what is f(2)?

Answer 1

The slope of the tangent line at (x,f(x)) is equivalent to df(x)/dx in other words the derivative of the function. We can find the actual function by taking the integral of df/dx as follows:

dy/dx = 3x+2

dy = 3x+2dx

∫dy = ∫3x+2 dx

y = 3x²/2 + 2x + C

then we know that the function passes through 10,5 so we can solve for C as follows:

5 = 3 * (10^2)/2 + 2(10) + C

5 = 3 * 50 + 20 + C

5 = 170 + C

C = -165

so the overall function is f(x) = 3x²/2 + 2x - 165

plugging in x = 2 we get that:

f(2) = 3(2^2)/2 + 2(2) - 165

f(2) = 6 + 4 - 165

f(2) = -155