Find the angle of intersection between the curves y = x²+2 & y=x+x^-1+1.
To find the angle of intersection between the curves y = x²+2 and y=x+x^(-1)+1, we can first find the points of intersection by solving the system of equations:
x² + 2 = x + x^(-1) + 1
Rearranging the equation:
x² - x - x^(-1) = -1
Multiplying both sides of the equation by x to eliminate the negative exponent:
x³ - x² - x = -x
Bringing all terms to one side of the equation:
x³ - x² - x + x - 1 = 0
Simplifying:
x³ - x² - 1 = 0
This is a cubic equation that cannot be easily factored. However, we can approximate the roots using numerical methods such as graphing or using a calculator.
Using a calculator or graphing software, we find that one of the roots is approximately x ≈ 1.32.
Now, we can substitute this value of x back into either of the original equations to find the corresponding y-value.
Using y = x² + 2:
y ≈ (1.32)² + 2 ≈ 3.74
Therefore, one of the points of intersection is (1.32, 3.74).
To find the angle of intersection between the curves, we can find the slopes of the tangents to each curve at the point of intersection.
For the curve y = x² + 2, the derivative is:
dy/dx = 2x
At the point (1.32, 3.74), the slope of the tangent line is:
m1 = 2(1.32) ≈ 2.64
For the curve y = x + x^(-1) + 1, the derivative is:
dy/dx = 1 - x^(-2)
At the point (1.32, 3.74), the slope of the tangent line is:
m2 = 1 - (1.32)^(-2) ≈ 1.23
The angle of intersection between the curves can be found using the formula:
tan(θ) = (m2 - m1) / (1 + m1*m2)
θ = arctan[(m2 - m1) / (1 + m1*m2)]
θ = arctan[(1.23 - 2.64) / (1 + 2.64*1.23)]
θ ≈ arctan[-1.4 / 4.2412]
Using a calculator, we find that the angle of intersection is approximately:
θ ≈ -18.6 degrees.