Find the point(s) on the curve y=x^2 that is/are closest to the point (0,3). If you use your calculator to do this (instead of taking a derivative), Sketch the graph that you use.
To find the point(s) on the curve y=x^2 that is closest to the point (0,3) without taking a derivative, we can use the distance formula between two points.
The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, (x1, y1) represents the point (0,3) and (x2, y2) represents a point on the curve y=x^2.
Let's substitute the values into the distance formula:
d = sqrt((x - 0)^2 + (x^2 - 3)^2)
Simplifying this expression, we have:
d = sqrt(x^2 + (x^2 - 3)^2)
Now, to find the point(s) where the distance is minimized, we need to find the value(s) of x that minimize the expression inside the square root.
If you use a calculator to graph this equation (d = sqrt(x^2 + (x^2 - 3)^2)) and find the minimum points, it will help to visualize the solution.
Here's how you can sketch the graph on a calculator:
1. Turn on your calculator and go to the graphing function.
2. Enter the equation: d = sqrt(x^2 + (x^2 - 3)^2)
3. Set the calculator to graph the function within a specific range of x-values that will capture the relevant portion of the graph (e.g., from -3 to 3).
4. Choose an appropriate scale for the x and y-axes to clearly see the shape of the plot.
5. Graph the equation and look for the point(s) where the distance is closest to (0,3). These will be the points where the graph is closest to the origin.
Once you have the graph, you can identify the x-values of the closest point(s) and then calculate the corresponding y-values by substituting those x-values back into the equation y = x^2.
I hope this helps! Let me know if there's anything else I can assist you with.