antiderivatives calculus

Question

If at any point (x, y) on the curve, d^2y/dx^2 = 24x + 10, and if there is a horizontal tangent line at the point (−1, 7), determine the equation of the curve.

Answer 1

We need to find the equation of the curve. The curve is simply going to be a function. If you took the derivative of that function at the point (-1, 7), then the derivative would be zero (since it has a horizontal tangent at this point). Also, if you took the derivative of the derivative (so second derivative) of this curve at any point, you would get 24x + 10 (thank goodness there is no y term in this equation ;). In a sense, we are starting at the end and working backwards. Since we have an equation for the second derivative of this curve, we need to take the antiderivative twice to get the function of the curve. Luckily, we are dealing with the power rule which is pretty straight forward. I am going to rewrite 24x +10 as 24x¹ + 10x⁰ (since x¹=x and x⁰=1). By the power rule, the antiderivative of 24x¹ is 24 x¹⁺¹ / (1+1) = 24x²/2 = 12x². The antiderivative of 10x⁰ is 10x⁰⁺¹/(0+1) = 10x¹/1 or just 10x. So, the antiderivative of 24x + 10 is 12x² + 10x + C. Why plus C? If you don't know, then don't worry about it, just remember to add C when taking the antiderivative. We just found the antiderivative of the second derivative, but what does that mean? The antiderivative of the second derivative is the first derivative. What do we know about the first derivative? We know dy/dx = 12x²+10x+C and also that dy/dx=0 at the point (-1,7). This means we can actually find C by plugging this information into the equation. Solve 0 = 12(-1)²+10(-1)+C for C and you should get -2. We are almost done. The antiderivative of the first derivative is the actual function (in this case, a curve). So, the antiderivative of dy/dx = 12x²+10x-2 is y = 4x³+5x²-2x + C. This time we can find C by plugging x = -1 and y = 7, since we know that is a point the curve passes through.