Final answer:
To find the stationary points of u = e^(-3x)tan(x), we differentiate the function, set the derivative equal to zero, and solve for x within the given interval. The derivative involves the product rule resulting in 0 = (-3e^(-3x)tan(x)) + (e^(-3x)sec^2(x)), and since e^(-3x) is never zero, we simplify to 0 = (-3tan(x)) + (sec^2(x)). We solve for x in the interval -1/2 < x < 1/2 using calculus and trigonometry.
Step-by-step explanation:
To find the x-coordinates of the stationary points of the curve with the equation u = e^(-3x)tan(x), we need to calculate the derivative of the function and set it equal to zero. A stationary point occurs where the derivative (slope) of the function is zero. As the derivative involves both the exponential and trigonometric functions, we will use the product rule to differentiate.
The derivative of u = e^(-3x)tan(x) is given by:
u' = (-3e^(-3x)tan(x)) + (e^(-3x)sec^2(x))
To find the stationary points, we set u' equal to zero:
0 = (-3e^(-3x)tan(x)) + (e^(-3x)sec^2(x))
Since e^(-3x) is never zero, we can simplify the equation to:
0 = (-3tan(x)) + (sec^2(x))
Now, we need to solve for x within the interval -1/2 < x < 1/2. We can use techniques from trigonometry and calculus to find the precise values for x that satisfy the equation.