Question

Найдите все значения х, при каждом из которых касательная к графику функции у=cos7x+7cosx в точках с абсциссой х параллельна касательной к этому же графику в точке с абсциссой пи/6

Answer 1

Производная у'=-7sin7x-7sinx...y'(п/6)=-7*(-0,5)-7*(0,5)=0-это тангенс угла наклона касательной во всех этих точках...решаем уравнение -7sin7x-7sinx=0...сокращаем на -7:sin7x+sinx=0...по формуле преобразования суммы в произведение получаем 2sin4x*cos3x=0...1).sin4x=0...4x=пк или х1=пк/4 (к принадл.Z)-1-й ответ....2).сos3x=0...3х=п/2+пк....х=п/6+пк/3 (к принадл.Z)-2-й ответ.

Answer 2

Finding the Values of x for Tangents Parallel to the Tangent at x = π/6

To find the values of x for which the tangent to the graph of the function y = cos(7x) + 7cos(x) is parallel to the tangent at x = π/6, we need to determine the slope of the tangent at x = π/6 and then find the values of x that have the same slope.

Let's start by finding the slope of the tangent at x = π/6. We can do this by taking the derivative of the function y = cos(7x) + 7cos(x) with respect to x.

The derivative of cos(7x) is -7sin(7x) (using the chain rule), and the derivative of 7cos(x) is -7sin(x) (using the chain rule). Therefore, the derivative of the function y = cos(7x) + 7cos(x) is:

y' = -7sin(7x) - 7sin(x)

Now, let's find the slope of the tangent at x = π/6 by substituting x = π/6 into the derivative:

y'(π/6) = -7sin(7(π/6)) - 7sin(π/6)

Simplifying this expression, we get:

y'(π/6) = -7sin(7π/6) - 7sin(π/6)

Using the trigonometric identity sin(π - θ) = sin(θ), we can simplify further:

y'(π/6) = -7sin(π/6) - 7sin(π/6)

y'(π/6) = -14sin(π/6)

y'(π/6) = -14(1/2)

y'(π/6) = -7

So, the slope of the tangent at x = π/6 is -7.

Now, we need to find the values of x for which the slope of the tangent is also -7. To do this, we can set the derivative equal to -7 and solve for x:

-7sin(7x) - 7sin(x) = -7

Dividing both sides of the equation by -7, we get:

sin(7x) + sin(x) = 1

Using the trigonometric identity sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2), we can rewrite the equation as:

2sin((7x + x)/2)cos((7x - x)/2) = 1

Simplifying further, we have:

2sin(4x)cos(3x) = 1

Dividing both sides of the equation by 2cos(3x), we get:

sin(4x)/cos(3x) = 1/2

Using the trigonometric identity tan(θ) = sin(θ)/cos(θ), we can rewrite the equation as:

tan(4x) = 1/2

To find the values of x that satisfy this equation, we can take the inverse tangent (arctan) of both sides:

4x = arctan(1/2)

x = (1/4)arctan(1/2)

Therefore, the values of x for which the tangent to the graph of the function y = cos(7x) + 7cos(x) is parallel to the tangent at x = π/6 are given by:

x = (1/4)arctan(1/2)

Please note that this is a general solution, and there may be additional values of x that satisfy the condition.