For the given function, find all points on an the graph where the tangent line is horizontal or vertical.
y= xsqrt(4-x^2)
y= sqrt(1-x^2)(4-x^2)
To find the points on the graph where the tangent line is horizontal or vertical, we need to find the points where the derivative of the function is either zero or undefined.
Let's begin with the first function: y = x√(4 - x^2).
First, we need to find the derivative of y with respect to x. Using the product rule, we get:
dy/dx = (d/dx)(x)(√(4 - x^2)) + (d/dx)(√(4 - x^2))(x)
= √(4 - x^2) + x(1/2)(-2x)(1/√(4 - x^2))
= √(4 - x^2) - x^2/√(4 - x^2)
= (√(4 - x^2))(1 - x^2/(4 - x^2))
To find where the tangent line is horizontal, we need to find the points where dy/dx = 0. Thus, we need to solve the equation (√(4 - x^2))(1 - x^2/(4 - x^2)) = 0.
Setting each factor equal to zero gives us two possibilities:
1. √(4 - x^2) = 0
This implies 4 - x^2 = 0
Solving for x, we find x = ±2.
2. 1 - x^2/(4 - x^2) = 0
This equation simplifies to 4 - x^2 = x^2
Here, we have 2x^2 = 4
Solving for x, we find x = ±√2.
Now, let's move on to the second function: y = √(1 - x^2)(4 - x^2).
As before, we need to find the derivative of y with respect to x. Using the product rule, we get:
dy/dx = (d/dx)(√(1 - x^2))(4 - x^2) + (d/dx)(4 - x^2)(√(1 - x^2))
= (√(1 - x^2))(2x) + (4 - x^2)(-x/√(1 - x^2))
= 2x√(1 - x^2) - x(4 - x^2)/√(1 - x^2)
= (2x√(1 - x^2) - 4x + x^3)/√(1 - x^2)
Again, to find where the tangent line is horizontal or vertical, we need to solve the equation dy/dx = 0 or where it is undefined.
1. (2x√(1 - x^2) - 4x + x^3)/√(1 - x^2) = 0
This equation can be rewritten as:
2x√(1 - x^2) - 4x + x^3 = 0
Unfortunately, solving this equation algebraically is quite complex. However, you can use numerical methods or graphing software to find approximate solutions.
2. For dy/dx to be undefined, the denominator √(1 - x^2) must be equal to zero:
√(1 - x^2) = 0
Solving for x, we find x = ±1.
In summary, for the first function y = x√(4 - x^2), the tangent line is horizontal at x = ±2 and vertical at x = ±√2. For the second function y = √(1 - x^2)(4 - x^2), finding the points where the tangent line is horizontal or vertical is more challenging and may require numerical methods or graphing software.