The slope of a curve is at the point (x,y) is 4x-3. Find the curve if it is required to pass through the point (1,1).

Work...
4(1)-3=1
y-1=1(x-1)
y=x

When they say "find the curve" they mean find the equation of function whose graph would be that curve.

your equation y = 2x^2 + 3x - 4 would graph to be a parabola, and a parabola is a curve.

As to the constant, remember that if you differentiate an equation like

y = 2x^2 + 3x - 4
you get y' = 4x + 3 - 0

so when we "anti-differentiate" that we really don't know what the value of the constant was, because its derivative would be zero no matter what the number was.
That is why we include a constant value of c or k to allow for that. Once we sub in the given point we then know the value of that constant

4x - 3 is your slope, so it is the derivative.

then dy/dx = 4x + 3, integrate to get

y = 2x^2 + 3x + k

plug in (1,1) into that to get k, and you are done!

why would I solve for the constant? I get k=-4. What is meant by curve?

To find the curve that passes through the point (1,1) with the given slope, we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.

In this case, the slope at the point (x,y) is given as 4x - 3, so we can substitute this into the equation: y - 1 = (4x - 3)(x - 1).

Expanding and simplifying the equation, we get: y - 1 = 4x^2 - 7x + 3.

To solve for y, we can move the constant terms to the other side: y = 4x^2 - 7x + 4.

Therefore, the curve that passes through the point (1,1) with the slope 4x - 3 is given by the equation y = 4x^2 - 7x + 4.