What is the equation of the curve that has gradient dy/dx=(4x-5) and passes through the point (3,7)?
To answer this question, we must use our knowledge of integration and differentiation.
First, we look at the information given to us. We have been told that the gradient of the curve is given by dy/dx=(4x-5). Since the gradient of the curve at a certain point is found by differentiating the equation of the curve, we can find the equation of the curve by integrating the differential. So, to get the equation we have to integrate dy/dx=(4x-5). Integrating this, we get y=2x2-5x+c.
We are still not finished. This last step is an important step that most people forget! We know that the curve passes through the point (3,7). To find the equation of the curve that satisfies this condition, we have to find the value of the constant c. To do so, we plug in the co-ordinates of the point that lies on the curve where x=3 and y=7. Doing this, we get the equation 7=2(3)2-5(3)+c. Rearranging this equation, we find out that c=7-18+15=4. Therefore, c=4 and the equation of the curve is y=2x2-5x+4.
