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Chapter 7: Problem 23

Find the points on the curve \(y=x+2 \cos (x)\) that have a horizontal tangent line.

Short Answer

Expert verified
The points on the curve \(y=x+2 \cos (x)\) that have a horizontal tangent line can be found by setting the derivative of the function equal to zero. They occur at \((\pi/6, \pi/6 + 2 \cos (\pi/6))\) and \((5\pi/6, 5\pi/6 + 2 \cos (5\pi/6))\), along with their coterminal angles.

Step by step solution

01

Calculate the derivative of the function

Start by taking the derivative of the given function. The derivative of \(y=x+2 \cos (x)\) is \(y'=1-2 \sin (x)\) using the Chain Rule.
02

Set the derivative equal to zero

Next, set the derivative equal to zero to find the x-values where the slope of the tangent line is zero. So solve the equation \(1-2 \sin (x) = 0\) for \(x\).
03

Solve for x

Rearrange the equation to get \(2 \sin (x) = 1\), so \(\sin (x) = 0.5\). The solutions to this equation are \(x = \pi/6 + 2\pi n\) and \(x = 5\pi/6 + 2\pi n\), where \(n\) is an integer.
04

Substitute \(x\) back into \(y\)

Finally, substitute the values of \(x\) back into the original function to find the values of \(y\). The points on the curve that have a horizontal tangent line will be \((\pi/6, \pi/6 + 2 \cos (\pi/6))\) and \((5\pi/6, 5\pi/6 + 2 \cos (5\pi/6))\), along with their coterminal angles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
Trigonometric functions, like sine and cosine, are fundamental in calculus and engineering.
They represent the relationships in right triangles and cyclical phenomena such as waves.
  • Cosine function, \cos(x), represents the x-coordinate of a point on the unit circle.
  • Sine function, \sin(x), is closely related, representing the y-coordinate.
The relationship \(\sin^2(x) + \cos^2(x) = 1\) is crucial, indicating the Pythagorean identity.
In our exercise, the cosine function appears in the expression \y = x + 2 \cos(x)\, necessitating the use of derivatives involving trigonometric identities.
Understanding these functions is key to solving problems involving periodic or oscillating behavior.
Horizontal Tangent Line
A horizontal tangent line occurs where the slope of the curve is zero.
This happens when the derivative of the function equals zero, meaning there's no vertical change.
  • Tangent lines provide a linear approximation of curves.
  • Horizontal tangents imply the curve is flat at those points.
In the problem, finding where \(1 - 2 \sin(x) = 0\) yields horizontal tangents.
Solving this gives critical values for \x\ which are where the curve flattens out, representing maximum, minimum, or saddle points in the context of broader analysis.
Chain Rule
The chain rule is a fundamental technique used to differentiate composite functions.
It is essential for handling expressions where one function is inside another.
  • The formula is \[ \( f(g(x)) \)' = f'(g(x)) \cdot g'(x) \]
  • This rule is crucial for working with nested functions like \cos(x)\.
In our derivative computation, \(y = x + 2 \cos(x)\), the chain rule simplifies the process of finding \(y' = 1 - 2 \sin(x)\).
It teaches us to take derivatives of inner functions first, followed by the derivatives of the outer functions, streamlining the approach to finding slopes.

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Most popular questions from this chapter

Find the derivative of \(\arccos \left(e^{x}\right)\)

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