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Chapter 16: Problem 84

At what angle does the graph of \(f(x)=\cos x\) depart from the point \((0,1)\) ?

Short Answer

Expert verified
The graph of the function \(f(x) = \cos x\) departs from the point \((0, 1)\) at an angle of \(0\) degrees or \(0\) radians.

Step by step solution

01

Differentiate the function

We have the function \(f(x) = \cos x\). To find the slope of the tangent line at any point, we need to differentiate this function with respect to x. \[f'(x) = \frac{d}{dx}(\cos x) = -\sin x\]
02

Find the slope at the given point

Now, we need to find the slope of the tangent line at the point \((0, 1)\). Since the function is given by \(f(x) = \cos x\), we have \(f(0) = \cos 0 = 1\). Thus, the given point lies on the graph of the function. Now we'll evaluate the derivative at \(x = 0\) to find the slope. \[f'(0) = -\sin(0) = 0\]
03

Find the angle of the tangent line

Since the slope of the tangent line is 0 at the point \((0, 1)\), it means that this tangent line is horizontal. Therefore, the angle at which the graph of the function departs from the point \((0, 1)\) is an angle of 0 degrees (or 0 radians). So the graph of the function \(f(x) = \cos x\) departs from the point \((0, 1)\) at an angle of 0 degrees or 0 radians.

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Key Concepts

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

Derivative
In calculus, the derivative of a function is a fundamental concept. It represents the rate at which a function is changing at any given point, often referred to as the "slope" of the function at that point. To find the derivative of a trigonometric function like \(f(x) = \cos x\), you apply rules such as the chain rule or simple differentiation rules for basic trigonometric functions.
  • For \(\cos x\), the derivative is \(-\sin x\), meaning at any point \(x\) on the function, the rate of change is given by \(-\sin x\).
  • The process of differentiation translates the geometric concept of a tangent line into the analytical calculation of the slope.
In the provided exercise, realizing that \(f'(x) = -\sin x\) reveals how \(\cos x\) can change direction along its curve. This understanding directly aids in determining how the function behaves around a specific point like \((0,1)\).
Tangent Line
A tangent line of a graph at any given point is a straight line that just touches the curve at that point and represents the immediate direction of the curve. The tangent does not cross the curve at the point of tangency, which means it gently "hugs" the curve.
  • To find the equation of the tangent line, you need the point and the slope. The slope is the derivative evaluated at the specific point.
  • In our exercise, at \((0,1)\) for \(f(x) = \cos x\), the derivative \(f'(0) = 0\) tells us the slope.
  • A slope of 0 indicates a horizontal tangent line, meaning the angle formed is 0 degrees or 0 radians from the horizontal.
Thus, the tangent line at \((0,1)\) for \(\cos x\) is horizontal, conveying that the function's direction at that precise moment is flat, not leaning upward or downward.
Trigonometric Functions
Trigonometric functions are cornerstone components in calculus. They help describe oscillating paths such as waves. Common trigonometric functions include \(\sin\), \(\cos\), and \(\tan\). In calculus, these functions often require differentiation to understand how they behave over their domain.
  • The function \(f(x) = \cos x\) is periodic and symmetrical, repeating every \(2\pi\) radians.
  • The rate of change for each trigonometric function is determined by its derivative, which is crucial for drawing tangent lines or understanding the behavior of its graph at specific points.
  • For the cosine function, this periodicity and its derivative \(-\sin x\) show how the curve inclines and declines.
Understanding trigonometric functions helps in grasping more complex calculus concepts such as Fourier series, which builds upon these periodic functions to solve real-world problems. Knowing how to differentiate these functions is vital for further exploration in calculus.

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