Consider the function f(x)=sqrt(x) and the point P(4,2) on the graph of f?
-Consider the graph f with secant lines passing through p(4,2) and Q(x,f(x)) for x-values 1, 3, and 5.
-Find the slope of each secant line
-Use the results to estimate the slope of the tangent line to the function at p(4,2). Describe how to improve your approximation of the slope?
I already graphed the function with the x values for 1, 3, and 5 but I'm confused on how to do the rest. Any help is greatly appreciated!
So far I tried to figure out the slope of each secant line:
For x=1 the slope is 1/3
For x=3 the slope is 2-sqrt(3)
For x=5 the slope is sqrt(5)-2
Any ideas?
the slopes look good.
The secant through (1,1) doesn't help much, but the lines through (3,√3) and (5,√5) bracket the tangent at (4,2).
Pick points closer to (4,2) to refine your value.
To find the slopes of the secant lines passing through point P(4,2) and the points Q(x,f(x)) for x-values 1, 3, and 5, we need to compute the slope of each line.
The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
slope = (y2 - y1) / (x2 - x1).
For each x-value (1, 3, and 5), we need to find the corresponding y-values by substituting them into the function f(x) = sqrt(x). Let's calculate the slopes for each secant line:
1. For x = 1:
- P(4,2) and Q(1, f(1)) = Q(1, sqrt(1))
- Slope = (sqrt(1) - 2) / (1 - 4)
2. For x = 3:
- P(4,2) and Q(3, f(3)) = Q(3, sqrt(3))
- Slope = (sqrt(3) - 2) / (3 - 4)
3. For x = 5:
- P(4,2) and Q(5, f(5)) = Q(5, sqrt(5))
- Slope = (sqrt(5) - 2) / (5 - 4)
Now, to estimate the slope of the tangent line to the function at P(4,2), we can consider the trend of the slopes of the secant lines as x approaches 4. As x-values get closer to 4, the secant lines approach the tangent line. Therefore, we can approximate the slope of the tangent line by taking the average of the slopes of the secant lines for x-values 3 and 5.
To improve the approximation of the slope, you can use smaller intervals of x-values around 4 and calculate the slopes of the corresponding secant lines. By taking smaller intervals, you will obtain more secant lines that get closer to the tangent line, providing a better estimate of the slope of the tangent line at P(4,2).