Let P(3,4) be a point on the circle x2 + y2 = 25.
calculate the Lim Mx as x approaches 3. How does this number relate to your answer in part b?
Part b the answer was to find the slope Mx of the line joining P and Q which is M=4-(root(25-x2))/3-x.
lim (x->3) 4-(√(25-x2))/(3-x)
Use l'Hospital's Rule to get
= lim (x->3) (x/√(25-x^2)]/(-1)
= (3/4)/(-1)
= -3/4
Recall that dy/dx at any point on the circle is -x/y
To find the limit of Mx as x approaches 3, we need to evaluate the expression for Mx at x = 3.
Given that P(3,4) is a point on the circle x^2 + y^2 = 25, we can substitute x = 3 into this equation to find the y-coordinate of point P.
3^2 + y^2 = 25
9 + y^2 = 25
y^2 = 25 - 9
y^2 = 16
y = ±√16
y = ±4
Since P(3,4) lies on the circle, the y-coordinate is 4, and the point P is (3,4).
Now let's calculate the slope Mx using the given formula:
Mx = [4 - √(25 - x^2)] / (3 - x)
Substituting x = 3 into this equation:
Mx = [4 - √(25 - 3^2)] / (3 - 3)
Mx = [4 - √(25 - 9)] / 0
Since the denominator is 0, the slope Mx is undefined at x = 3.
Therefore, the limit of Mx as x approaches 3 does not exist.
This result is different from the answer in part b because the slope Mx cannot be calculated using the given formula at x = 3 due to the division by 0.
To calculate the limit of Mx as x approaches 3, we need to evaluate the slope Mx of the line joining P(3, 4) and Q(x, y) as x approaches 3.
In part b, we found the formula for Mx, which is given by Mx = (y - 4) / (x - 3). To find the slope Mx, we need to find the value of y corresponding to the given x-coordinate.
Given the equation of the circle x^2 + y^2 = 25, we substitute x = 3 into this equation to solve for y.
3^2 + y^2 = 25
9 + y^2 = 25
y^2 = 25 - 9
y^2 = 16
y = ±√16
y = ±4
Since P(3, 4) lies on the upper half of the circle, the value of y we choose is y = 4.
Substituting the values of x and y into the formula for Mx, we get:
Mx = (4 - 4) / (x - 3) = 0 / (x - 3) = 0
As x approaches 3, the value of Mx approaches 0. Therefore, the limit of Mx as x approaches 3 is 0.
This means that the line joining P(3, 4) and any point Q on the circle (x^2 + y^2 = 25) becomes steeper as Q moves closer to the point P. However, when Q reaches the point P itself, the slope becomes 0.
In summary, the limit of Mx as x approaches 3 is 0, indicating that the slope of the line joining P and Q becomes steeper as Q approaches P but becomes 0 when Q coincides with P.