How do you find the x-coordinates of all points on the curve #y=sin2x-2sinx# at which the tangent line is horizontal?

Answer 1

In the interval #0 ≤ x ≤ 2pi#, the points are #x= (5pi)/6, (7pi)/6 and 0#.

The slope of the tangent where it is horizontal is #0#. Since the output of the derivative is the instantaneous rate of change (slope) of the function, we will need to find the derivative.
#y = sin2x - 2sinx#
Let #y = sinu# and #u = 2x#. By the chain rule, #dy/dx= cosu xx 2 = 2cos(2x)#.
#y' = 2cos(2x) - 2cosx#
We now set #y'# to #0# and solve for #x#.
#0 = 2cos(2x) - 2cosx#
Apply the identity #cos(2beta) = 2cos^2beta - 1#
#0 = 2(2cos^2x - 1) - 2cosx#
#0 = 4cos^2x - 2cosx - 2#
#0 = 2(2cos^2x - cosx - 1)#
#0 = 2cos^2x - 2cosx + cosx - 1#
#0 = 2cosx(cosx- 1) + 1(cosx - 1)#
#0= (2cosx + 1)(cosx - 1)#
#cosx= -1/2 and cosx = 1#
#x = (5pi)/6, (7pi)/6 and 0#

Hopefully this helps!

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the x-coordinates of all points on the curve y = sin(2x) - 2sin(x) at which the tangent line is horizontal, we need to find the values of x where the derivative of the function is equal to zero.

First, let's find the derivative of y with respect to x:

dy/dx = d/dx(sin(2x) - 2sin(x)) = 2cos(2x) - 2cos(x)

Next, we set the derivative equal to zero and solve for x:

2cos(2x) - 2cos(x) = 0

Now, we can simplify this equation:

cos(2x) - cos(x) = 0

Using the trigonometric identity cos(2x) = 2cos^2(x) - 1, we can rewrite the equation as:

2cos^2(x) - 1 - cos(x) = 0

Rearranging the terms:

2cos^2(x) - cos(x) - 1 = 0

Now, we can solve this quadratic equation for cos(x) using factoring, quadratic formula, or other methods. Once we find the values of cos(x), we can find the corresponding values of x by taking the inverse cosine (arccos) of each cos(x) value.

These x-values will represent the x-coordinates of all points on the curve where the tangent line is horizontal.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To find the x-coordinates of all points on the curve ( y = \sin(2x) - 2\sin(x) ) at which the tangent line is horizontal, we need to find where the derivative of the function with respect to x is equal to zero, as the derivative gives us the slope of the tangent line.

First, let's find the derivative of the function ( y = \sin(2x) - 2\sin(x) ) with respect to x:

[ \frac{dy}{dx} = 2\cos(2x) - 2\cos(x) ]

Now, we set the derivative equal to zero and solve for x:

[ 2\cos(2x) - 2\cos(x) = 0 ]

[ \cos(2x) = \cos(x) ]

[ 2x = 2n\pi \pm x ]

[ x = n\pi ]

So, the x-coordinates of all points on the curve where the tangent line is horizontal are ( x = n\pi ), where ( n ) is any integer.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7