Find the equation of the tangent to the curve y = 2x^2 + x - 1 at the point where x = 1.
To get the required tangent equation we need its gradient and the coordinates of a point it passes through. We can then substitute it into the formula y - y1 = m(x - x1). 1. Finding the gradient: To find the gradient of the tangent, first find the gradient of the curve where x=1: y = 2x2 + x - 1 --> dy/dx = 4x + 1 When x = 1, dy/dx= 4(1)+1 = 4+1 = 5 So the gradient of the required tangent is 5. 2. Finding a point the tangent passes through: We know that the tangent touches the curve when its x coordinate is 1, and we can also calculate the y coordinate of this point: y = 2(1)2 + (1) - 1 = 2 (as y = 2x2 + x - 1). 3. Substiuting this into the line equation formula and rearranging: We can use the line equation formula: y - y1 = m(x - x1) and subsitute in the information we now have (m=5, x1=1 and y1=2). y - 2 = 5(x - 1) y - 2 = 5x - 5 --> y = 5x - 3 So y = 5x - 3 is the equation of the tangent to the curve y = 2x2 + x - 1 at the point where x = 1.
