consider the curve defined by the equation y=a(x^2)+bx+c. Take a point(h,k) on the curve. use Wallis's method of tangents to show that the slope of the line tangent to this curve at the point(h,k) will be m= 2ah+b. have to prove this for tow cases: a>0 and a<0.
Thank you
To show that the slope of the tangent line to the curve defined by the equation y = ax^2 + bx + c at the point (h, k) is given by m = 2ah + b, we will use Wallis's method of tangents.
Wallis's method of tangents involves finding the equation of the tangent line at the given point and then calculating its slope.
Let's start by finding the equation of the tangent line in general form, y - k = m(x - h), where (h, k) is the given point and m is the slope of the tangent line.
1. Case: a > 0
Assuming a > 0, we can proceed as follows:
Step 1: Calculate the first derivative of the curve.
Differentiating y = ax^2 + bx + c with respect to x, we get dy/dx = 2ax + b.
Step 2: Find the slope of the tangent line.
Substituting the x-coordinate of the given point into the derivative expression, we have:
m = dy/dx at x = h = 2ah + b. This gives us the slope of the tangent line.
Therefore, for the case where a > 0, the slope of the tangent line at the point (h,k) is m = 2ah + b.
2. Case: a < 0
Now, let's consider the case where a < 0:
Step 1: Calculate the first derivative of the curve.
Differentiating y = ax^2 + bx + c with respect to x, we get dy/dx = 2ax + b.
Step 2: Find the slope of the tangent line.
Substituting the x-coordinate of the given point into the derivative expression, we have:
m = dy/dx at x = h = 2ah + b. This gives us the slope of the tangent line.
Therefore, for the case where a < 0, the slope of the tangent line at the point (h,k) is m = 2ah + b.
In both cases, a > 0 and a < 0, the slope of the tangent line is m = 2ah + b, as derived using Wallis's method of tangents.