Question

1.       Find the function whose tangent has slope 4x + 1 for each value of x and whose graph passes through the point (1, 2).

Answer 1

Let F(x) be the unknown function. Since tangent slope is equal to derivative, then for each x

$F'(x)=4x+1\implies F(x)=\intop (4x+1)dx=2x^2+x+C$

As we know, the graph of that function passes through point (1;2), then F(1) = 2

$F(1)=2*1^2+1+C=3+C=2\implies C=-1$

$F(x)=2x^2+x-1$ is the required function.