airplanes fly from left to right along the path y=1+(1/x)and can shoot their bullets in the tangent direction at creatures placed along the x-axis at x=1,2,3,4 and 5.determine whether a creature will be hit if the player shoots when the plane is at p(1,2)
dy/dx = slope = -1/x^2
at (1,2)
y = 2 and slope = -1
where does that hit the x axis
at x = 2 it is at y = 2-1 = 1
at x = 3 it is at y = 1-1 = 0
so it hits anything at (3,0)
To determine whether a creature will be hit when the plane is at point P(1,2), we first need to find the equation of the tangent line to the curve y = 1 + (1/x) at point P(1,2). Then we can check if any of the x-coordinate values of the creatures (1, 2, 3, 4, 5) lie on or below the tangent line. If any of them do, it means the plane's bullets will hit a creature.
Let's begin by finding the equation of the tangent line at point P(1,2).
First, we need to find the derivative of the curve y = 1 + (1/x) with respect to x. Taking the derivative will give us the slope of the tangent line at any point on the curve.
dy/dx = d/dx (1 + (1/x))
= 0 + (-1/x^2)
= -1/x^2
Now that we have the derivative, we can find the slope of the tangent line at point P(1,2) by substituting x = 1 into the derivative equation:
m = -1/(1^2)
= -1/1
= -1
So, the slope of the tangent line at P(1,2) is -1.
Next, we can use the point-slope form of a linear equation to find the equation of the tangent line. The point-slope form is given by:
y - y1 = m(x - x1)
Where (x1, y1) is the given point and m is the slope of the line.
Substituting P(1,2) and the slope -1 into the equation, we get:
y - 2 = -1(x - 1)
y - 2 = -x + 1
Rearranging the equation in slope-intercept form, we have:
y = -x + 3
Now, we need to check if any of the creatures' x-coordinate values (1, 2, 3, 4, 5) lie on or below the tangent line y = -x + 3.
Substituting each x value into the equation, we can find the corresponding y value:
For x = 1: y = -(1) + 3 = 2
For x = 2: y = -(2) + 3 = 1
For x = 3: y = -(3) + 3 = 0
For x = 4: y = -(4) + 3 = -1
For x = 5: y = -(5) + 3 = -2
From the calculations, we can see that the y-values for creatures at x = 1 and x = 2 are both above the tangent line, y = -x + 3. However, the y-values for creatures at x = 3, x = 4, and x = 5 are all below the tangent line.
Therefore, only the creatures at x = 3, x = 4, and x = 5 will be hit if the player shoots when the plane is at point P(1,2).