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.
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To show that the slope of the line tangent to the curve y = ax^2 + bx + c at the point (h, k) is m = 2ah + b, we will use Wallis's method of tangents. This method involves finding the equation of the tangent line at the point (h, k) and then determining its slope.
1) Case: a > 0
First, let's find the equation of the tangent line at the point (h, k). The tangent line will have the form y = mx + d, where m represents the slope of the line and d is a constant.
Since the line is tangent to the curve at (h, k), it means that the point (h, k) lies on the line. Plugging these values into the equation, we have k = mh + d.
Now, let's differentiate the equation of the curve with respect to x: y = ax^2 + bx + c.
dy/dx = 2ax + b.
To find the slope of the tangent line at the point (h, k), we substitute x = h into the derivative:
m = dy/dx |(h) = 2ah + b.
Therefore, the slope of the line tangent to the curve y = ax^2 + bx + c at the point (h, k) when a > 0 is m = 2ah + b.
2) Case: a < 0
Using the same process as above, we start with the equation of the curve y = ax^2 + bx + c.
Differentiating the equation with respect to x, we have dy/dx = 2ax + b.
Again, we substitute x = h into the derivative:
m = dy/dx |(h) = 2ah + b.
Therefore, the slope of the line tangent to the curve y = ax^2 + bx + c at the point (h, k) when a < 0 is m = 2ah + b.
In both cases, we have shown that the slope of the line tangent to the curve at the point (h, k) is m = 2ah + b using Wallis's method of tangents.