Given the curve defined by the equation y=cos^2(x) + sqrt(2)* sin(x) with domain (0,pi) , find all points on the curve where the tangent line to the curve is horizontal
hint:
solve for dy/dx=0 on the interval [0,π].
y=cos^2(x) + sqrt(2)* sin(x)
dy/dx=-2cos(x)sin(x)+(√2)cos(x) = 0
To find the points on the curve where the tangent line is horizontal, we need to find the values of x that make the derivative equal to zero. The derivative of the curve equation, y = cos^2(x) + sqrt(2)*sin(x), will give us the slope of the tangent line at each point.
Step 1: Find the derivative of the function y with respect to x.
To find the derivative, we differentiate each term separately using the rules of differentiation. Note that we can simplify the equation by using the trigonometric identity cos^2(x) = 1/2 + (1/2)*cos(2x).
dy/dx = d/dx (1/2 + (1/2)*cos(2x) + sqrt(2)*sin(x))
= 0 - sin(2x) + sqrt(2)*cos(x)
Step 2: Find the values of x where the derivative is equal to zero.
To find these points, set dy/dx equal to zero and solve for x.
0 - sin(2x) + sqrt(2)*cos(x) = 0
sin(2x) = sqrt(2)*cos(x)
Step 3: Solve for x using trigonometric identities and equations.
We can use the trigonometric identity sin(2x) = 2*sin(x)*cos(x) to rewrite the equation as:
2*sin(x)*cos(x) = sqrt(2)*cos(x)
Dividing both sides by cos(x), assuming cos(x) ≠ 0, we get:
2*sin(x) = sqrt(2)
sin(x) = sqrt(2)/2
Now, we need to find the values of x in the given domain (0, pi) that satisfy the above equation. The sine function is positive in the first and second quadrants, which means we need to find the angles whose sine is sqrt(2)/2.
Using the unit circle or trigonometric identities, we can determine that the angles are π/4 and 3π/4.
Step 4: Find the corresponding y-values.
Now that we have the x-values where the derivative is zero, we can substitute these values into the original equation y = cos^2(x) + sqrt(2)*sin(x) to find the corresponding y-values.
For x = π/4:
y = cos^2(π/4) + sqrt(2)*sin(π/4)
Simplifying using trigonometric identities:
y = (1/2) + sqrt(2)*(1/2)
y = 1 + sqrt(2)/2
For x = 3π/4:
y = cos^2(3π/4) + sqrt(2)*sin(3π/4)
Simplifying using trigonometric identities:
y = (1/2) + sqrt(2)*(-1/2)
y = 1 - sqrt(2)/2
So, the two points on the curve where the tangent line is horizontal are (π/4, 1 + sqrt(2)/2) and (3π/4, 1 - sqrt(2)/2).