The point P(4,2) lies on the curve y=x^1/2
If Q is the point (x,x^1/2) use your calculator to find the slope of the secant line PQ for the following values of X:
1)3.5
2)3.9
3)3.99
4)3.999
5)4.5
6)4.1
7)4.01
8)4.001
i plugged these points into the intial equation and got that it was approaching 2. and i made it the slope of secant. however, my friend got a different answer; she found the slope using delta y over delta x. who is right and why?
thanks for any help!
4) If the coordinates of Q are
x = 3.999, y = 1.99975
then the slope of the "secant" line between Q and P (4,2) is
(.00025)/.001 = 0.2500 = 1/4
The exact value of the slope of the tangent to the y(x) curve at P is dy/dx = (1/2)/sqrt x = 1/4
For points Q farther from P, you will get a slightly different answer
To find the slope of the secant line PQ, we need to calculate the change in y-values divided by the change in x-values between the points P and Q for each given x-value.
Your method of plugging in the points into the equation, y = x^(1/2), to find the corresponding y-values is correct. However, using the formula "delta y over delta x" (or rise over run) is a more systematic approach that applies to any function.
The steps to find the slope using "delta y over delta x" are as follows:
1) Calculate the difference in y-values (delta y) by subtracting the y-coordinate of P from the y-coordinate of Q.
2) Calculate the difference in x-values (delta x) by subtracting the x-coordinate of P from the x-coordinate of Q.
3) Divide the delta y by delta x to get the slope of the secant line.
Let's find the slope of the secant line PQ for each given value of x:
1) For x = 3.5:
- Delta y = 3.5^(1/2) - 2
- Delta x = 3.5 - 4
- Slope = (3.5^(1/2) - 2) / (3.5 - 4)
2) For x = 3.9:
- Delta y = 3.9^(1/2) - 2
- Delta x = 3.9 - 4
- Slope = (3.9^(1/2) - 2) / (3.9 - 4)
3) For x = 3.99:
- Delta y = 3.99^(1/2) - 2
- Delta x = 3.99 - 4
- Slope = (3.99^(1/2) - 2) / (3.99 - 4)
4) For x = 3.999:
- Delta y = 3.999^(1/2) - 2
- Delta x = 3.999 - 4
- Slope = (3.999^(1/2) - 2) / (3.999 - 4)
5) For x = 4.5:
- Delta y = 4.5^(1/2) - 2
- Delta x = 4.5 - 4
- Slope = (4.5^(1/2) - 2) / (4.5 - 4)
6) For x = 4.1, 7) For x = 4.01, and 8) For x = 4.001:
- Repeat the above steps by using the provided values of x.
By calculating the slopes using either method, you and your friend should obtain the same values. Both methods are valid, but using "delta y over delta x" is a more general approach that can be used for any function and helps to establish a consistent method for finding the slope of a secant line.