Question

Найти значение параметра а , при котором касательная к графику ф-ии у=а×sinх+а в точке с абсциссой х =п/6 параллельна прямой у=х . Написать ур-е касательной

Answer 1

Y=a*sinx    x0=π/6   y'=acosx    y'(π/6)=a*cosπ/6=a√3/2
из параллельности с прямой у=1*х следует равенство угловых коэффициентов  a√3/2=1   a=2/√3=2√3/3

y(π/6)=asinπ/6+a=1/2*a+a=3/2*a=3/2*2/3*√3=√3
y'(π/6)=a√3/2=2√3*√3/3*2=6/6=1
уравнение касательной 1*(х-π/6)+√3=х+√3-π/6

Answer 2

Finding the Value of Parameter 'a' and Writing the Equation of the Tangent Line

To find the value of the parameter 'a' and write the equation of the tangent line to the graph of the function f(x) = a * sin(x) at the point with x-coordinate x = π/6 that is parallel to the line y = x, we can follow these steps:

1. Determine the derivative of the function f(x) = a * sin(x) using the chain rule. The derivative of sin(x) is cos(x), and the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Therefore, the derivative of f(x) = a * sin(x) is f'(x) = a * cos(x).

2. Find the value of 'a' that makes the derivative f'(x) = a * cos(x) equal to the slope of the line y = x. Since the line y = x has a slope of 1, we set f'(x) equal to 1 and solve for 'a': ``` a * cos(x) = 1 ```

3. Substitute x = π/6 into the equation a * cos(x) = 1 to find the value of 'a': ``` a * cos(π/6) = 1 ```

The value of cos(π/6) can be determined using trigonometric identities. Cosine of π/6 is equal to √3/2. Therefore, we have: ``` a * (√3/2) = 1 ```

Solving for 'a', we get: ``` a = 2/√3 ```

4. Now that we have the value of 'a', we can write the equation of the tangent line. The equation of a line can be written in the point-slope form, which is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope of the line.

The point (x₁, y₁) is (π/6, f(π/6)) since we want the tangent line to pass through the point with x-coordinate x = π/6 on the graph of f(x) = a * sin(x).

Substituting the values, we have: ``` (y - f(π/6)) = (a * cos(π/6))(x - π/6) ```

Since we know the value of 'a' is 2/√3 and cos(π/6) is √3/2, we can simplify the equation to: ``` (y - f(π/6)) = (2/√3)(√3/2)(x - π/6) ```

Simplifying further, we get: ``` (y - f(π/6)) = (x - π/6) ```

Therefore, the equation of the tangent line to the graph of f(x) = a * sin(x) at the point with x-coordinate x = π/6 that is parallel to the line y = x is: ``` y - f(π/6) = x - π/6 ```

Please note that the exact value of f(π/6) depends on the value of 'a'.