Problem 30 $29-30=$ Show that $f$ is co... [FREE SOLUTION]

Chapter 1: Problem 30

$29-30=$ Show that $f$ is continuous on $(-\infty, \infty)$ $$f(x)=\left\\{\begin{array}{ll}{\sin x} & {\text { if } x<\pi / 4} \\ {\cos x} & {\text { if } x \geqslant \pi / 4}\end{array}\right.$$

Short Answer

Expert verified

The function $f(x)$ is continuous for all $x$.

Step by step solution

Analyze Continuity for x < 蟺/4

For the interval where $x < \frac{\pi}{4}$, the function $f(x) = \sin x$ is continuous. This is because $\sin x$ is continuous everywhere on $(-fty, \infty)$. Therefore, for every point in this range, the left-hand limit and the function value are the same.

Analyze Continuity for x 鈮� 蟺/4

For the interval where $x \geq \frac{\pi}{4}$, the function $f(x) = \cos x$ is again continuous. The $\cos x$ function is continuous everywhere on $(-fty, \infty)$. Thus, for every point in this range, the right-hand limit and the function value are the same.

Check Continuity at the Transition Point x = 蟺/4

Now, examine the point $x = \frac{\pi}{4}$, where the definition of the function changes. Compute the left-hand limit approaching $\frac{\pi}{4}$ from below, which is $\lim_{{x \to \frac{\pi}{4}^-}} \sin x = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$. Compute the right-hand limit approaching from above, which is $\lim_{{x \to \frac{\pi}{4}^+}} \cos x = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$. Since the left-hand and right-hand limits are equal, and $f(\frac{\pi}{4}) = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, the function is continuous at $x = \frac{\pi}{4}$.

Conclusion

Since $f(x)$ is continuous for $x < \frac{\pi}{4}$, continuous for $x \geq \frac{\pi}{4}$, and continuous at $x = \frac{\pi}{4}$, the function $f(x)$ is continuous on $(-fty, \infty)$.

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Key Concepts

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

Continuity at a Point

Continuity at a point is about making sure a function behaves smoothly around that specific location. For a function to be continuous at a particular point, three conditions must be met:

The function must be defined at the point.
The left-hand and right-hand limits of the function, as they approach the point, must exist.
The left-hand and right-hand limits must be equal to the function's value at that point.

When these conditions are satisfied, there are no jumps, breaks, or holes at that point. This ensures that as x approaches any particular value, the function approaches exactly the value you'd expect it to at that point. In the context of our function, this means checking its behavior around where the function switches from $\sin x$ to $\cos x$, especially at the transition point $x = \frac{\pi}{4}$.

Sine and Cosine Functions

Sine and cosine functions are both foundational trigonometric functions known for their smooth, wave-like patterns.

The function $\sin x$ oscillates between -1 and 1, producing a wave that starts at 0 when $x = 0$, rises to 1 at $x = \frac{\pi}{2}$, falls to 0 at $x = \pi$, and continues this pattern as x increases.
The function $\cos x$, similarly, fluctuates between -1 and 1, but begins at 1 when $x = 0$, reaches 0 at $x = \frac{\pi}{2}$, hits -1 at $x = \pi$, and repeats its cycle.

Both functions are continuous over the entire real line, meaning there are no breaks in their waves. This continuity is crucial for proving the overall continuity of functions that incorporate $\sin x$ and $\cos x$, like our piecewise function.

Left-Hand and Right-Hand Limits

Left-hand and right-hand limits are tools used to assess behavior of functions as they approach specific points from different directions.

A left-hand limit, $\lim_{{x \to c^-}} f(x)$, is how a function approaches a value as x approaches the point c from the left (less than c).
A right-hand limit, $\lim_{{x \to c^+}} f(x)$, is how a function approaches a value as x approaches that same point from the right (greater than c).

These limits are especially important for piecewise functions at transition points where the definition of the function changes. In our function's case, at $x = \frac{\pi}{4}$, both limits must be calculated. The left-hand limit of $\sin x$ and the right-hand limit of $\cos x$ both lead to $\frac{\sqrt{2}}{2}$, showing continuity at this critical point.

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91影视

\(29-30=\) Show that \(f\) is continuous on \((-\infty, \infty)\) $$f(x)=\left\\{\begin{array}{ll}{\sin x} & {\text { if } x<\pi / 4} \\ {\cos x} & {\text { if } x \geqslant \pi / 4}\end{array}\right.$$

Short Answer

Step by step solution

Analyze Continuity for x < 蟺/4

Analyze Continuity for x 鈮� 蟺/4

Check Continuity at the Transition Point x = 蟺/4

Conclusion

Key Concepts

Continuity at a Point

Sine and Cosine Functions

Left-Hand and Right-Hand Limits

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