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Chapter 2: Problem 13

Let \(f(x)=\left\\{\begin{array}{l}\sin \pi x,-11\end{array} \quad\right.\) Plot \(f(x)\) and \(f^{\prime}(x)\) on the interv: \({[-5,5] . }\)

Short Answer

Expert verified
Plot \( \sin (\pi x) \) for \(-1 < x \leq 1\) and \( \frac{1}{2}x^2\) for \( x > 1 \) with their derivatives \( \pi \cos (\pi x)\) and \( x \), noting the discontinuity at \( x = 1 \).

Step by step solution

01

- Define the Piecewise Function

The function is defined differently for two intervals of x. For \(-1 < x \leq 1\), \(f(x) = \sin (\pi x)\). For \(x > 1\), \(f(x) = \frac{1}{2}x^{2}\).
02

- Calculate the Derivative for Each Interval

Differentiate each piece of the function. For \( -1 < x \leq 1 \),\( f'(x) = \pi \cos (\pi x)\). For \( x > 1 \), \( f'(x) = x \).
03

- Analyze Continuity at the Transition Point

Check the value of the original function and its derivative at \( x = 1 \) to ensure smooth transition. \( f(1) = \sin (\pi) = 0\) and \( \lim_{{x \to 1^+}} \frac{1}{2} x^2 = \frac{1}{2} (1)^2 = \frac{1}{2} \). Hence, there is a discontinuity in \(f\) and \(f'\).
04

- Plot the Function \( f(x) \)

Create a graph of the piecewise function over the interval \([-5, 5]\). For \( -1 < x \leq 1\), plot \( \sin (\pi x) \), and for \( x > 1\), plot \( \frac{1}{2}x^{2} \). Don't forget to mark the discontinuity.
05

- Plot the Derivative \( f'(x) \)

Graph \( f'(x) \) on the interval \([-5, 5]\). For \( -1 < x \leq 1 \), plot \( \pi\cos (\pi x) \), and for \( x > 1 \), plot \( x \). Again, mark the discontinuity in the derivative.

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Key Concepts

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

Derivative Calculation
When calculating the derivative of a piecewise function, it鈥檚 important to evaluate each segment separately. For the function given, we start by identifying the segments:
\(-1 < x \leq 1\) where \(f(x) = \sin (\pi x)\)
and
\(x > 1\) where \(f(x) = \frac{1}{2}x^2\).
Next, we find the derivatives:
  • For \(-1 < x \leq 1\), the derivative of \(f(x)\) is \(f'(x) = \pi \cos (\pi x)\).

  • For \(x > 1\), the derivative of \(f(x)\) is \(f'(x) = x\)
This shows how the function鈥檚 rate of change varies over different intervals. Be meticulous in differentiating each piece correctly.
Continuity Analysis
Checking for continuity in piecewise functions requires examining transition points. For our given function, consider the transition at \(x = 1\). We need to ensure that the function value and the derivative value match at this point:
  • \(f(1) = \sin (\pi) = 0\)
  • \( \lim_{{x \to 1^+}} \frac{1}{2} x^2 = \frac{1}{2}*1^2 = \frac{1}{2} \)
Since \(f(1)\) and \( \lim_{{x \to 1^+}} f(x)\) are different, we observe a discontinuity. Also, verify the derivatives:
  • \(f'(1) = \pi \cos (\pi) = \pi\)

  • \( \lim_{{x \to 1^+}} f'(x) = 1 \)
Again, differing values indicate a discontinuity in the derivative at \(x = 1\).
Mathematical Graphing
Graphing piecewise functions requires plotting each segment separately. For this function, focus on the interval from \([-5, 5]\):
  • Plot \( \sin (\pi x)\) from \(-1 < x \leq 1\)
  • Plot \( \frac{1}{2} x^2\) for \(x > 1\)
Clearly mark where different segments join, and highlight any discontinuities. For our example, the function jumps at \(x = 1\). Graphing the derivative similarly involves:
  • Plot \( \pi \cos (\pi x)\) for \(-1 < x \leq 1\)

  • Plot \(x\) for \(x > 1\)
Carefully draw these to show the change in rate and mark the discontinuity in the derivative. This visual representation helps understand behavior and ensures checking for mistakes.
Discontinuities in Functions
Discontinuities occur when a function jumps or breaks at certain points. For piecewise functions, this is often at the boundaries between pieces. In our function, the value and derivative at \(x = 1\) are key:
  • Function jump: \(f(1) = 0\) versus \( \lim_{{x \to 1^+}} f(x) = \frac{1}{2} \)
  • Derivative jump: \(f'(1) = \pi\) versus \( \lim_{{x \to 1^+}} f'(x) = 1 \)
These differing values confirm discontinuities. Note that discontinuities can be:
  • Removable if function can be redefined at the point
  • Jump when function abruptly changes value
  • Infinite when values go toward infinity
  • Oscillatory if function rapidly flips between values
Understanding these helps better analyze and accurately interpret piecewise functions and their graphs.

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Most popular questions from this chapter

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