Problem 1 - Calculus for Biology and Medicine

Question

Let $$f(x)=x^{2} \cos \frac{1}{x}, \quad x \neq 0$$ (a) Use a graphing calculator to sketch the graph of \(y=\overline{f(x) \text { . }}\) (b) Show that $$-x^{2} \leq x^{2} \cos \frac{1}{x} \leq x^{2}$$ holds for \(x \neq 0\) (c) Use your result in (b) and the sandwich theorem to show that $$ \lim _{x \rightarrow 0} x^{2} \cos \frac{1}{x}=0 $$

Step-by-Step Solution

Answer

(a) Sketch shows oscillation shrinking towards zero. (b) Inequality holds since \(-1 \leq \cos\frac{1}{x} \leq 1\). (c) Limit is zero by the sandwich theorem.

Step 1: Sketch the Graph

To sketch the graph of \(y = f(x) = x^2 \cos \frac{1}{x}\), use a graphing calculator. Input \(x^2 \cos \frac{1}{x}\) as the function. Observe that as \(x\) approaches zero, the oscillations become more rapid, and the amplitude, dictated by \(x^2\), approaches zero. The graph should show oscillation between \(y = -x^2\) and \(y = x^2\).

Step 2: Establish Inequality Boundaries

To show \(-x^2 \leq x^2 \cos \frac{1}{x} \leq x^2\), note that the cosine function satisfies \(-1 \leq \cos \frac{1}{x} \leq 1\) for all \(x eq 0\). Multiplying through by \(x^2\) (which is non-negative), gives the inequality \(-x^2 \leq x^2 \cos \frac{1}{x} \leq x^2\).

Step 3: Apply the Sandwich Theorem

The sandwich theorem states that if \(g(x) \leq f(x) \leq h(x)\) and \(\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L\), then \(\lim_{x \to a} f(x) = L\). In our case, \(-x^2 \leq x^2 \cos \frac{1}{x} \leq x^2\) and both \(\lim_{x \to 0} -x^2 = 0\) and \(\lim_{x \to 0} x^2 = 0\). Hence, by the sandwich theorem, \(\lim_{x \to 0} x^2 \cos \frac{1}{x} = 0\).

Key Concepts

Sandwich Theorem

The Sandwich Theorem is a handy tool in calculus. It simplifies finding limits of oscillating functions. Imagine trying to pin down a value by using two other functions. Here's how it works: - If you can find two functions that "squeeze" your function from above and below, you can use their limits. - If both boundary functions converge to the same limit at a point, then the function "sandwiched" between them must also converge to that limit. In practical terms, you often encounter it with equations or inequalities. With the right inequalities, the theorem helps establish difficult limits neatly and elegantly. This technique works especially well for functions that oscillate or behave unpredictably near a specific point.

Limit

A limit provides a way to determine how a function behaves as it approaches a specific point. Think of it as predicting where the "end" of a function is going as you get closer to a particular value of x. For example, consider what happens to the value of function f(x) = x^2 cos(1/x) as x approaches 0. - Although the function might look erratic due to the oscillating cosine part, using the limit helps identify the value towards which the function converges. - In many cases, you evaluate limits where direct substitution is difficult. Instead, you rely on intuitive understanding and mathematical rules like the Sandwich Theorem. In this specific example, despite the oscillations, using the inequalities from the Sandwich Theorem helps show that the limit of f(x) as x approaches 0 is indeed 0.

Graphing Calculator

Graphing calculators are powerful tools that aid in visualizing functions. When trying to understand complex functions, like f(x) = x^2 cos(1/x), sketches can reveal details not easily discernible otherwise. Some benefits of using a graphing calculator include:

• Providing a visual sense of function behavior, including peaks, valleys, and oscillations.
• Allowing for manipulation of the view, zooming into areas where the function acts unpredictably.
• Helping confirm mathematical hypotheses graphically before delving into algebraic proofs.

For this specific function, you would see the graph oscillating with increasingly narrow margins. As x tends towards 0, the amplitude of the oscillations decreases, surrounding the x-axis.

Inequality Boundaries

In mathematics, establishing inequality boundaries is crucial. They help us understand how a function is constrained. For functions like f(x) = x^2 cos(1/x), the key is to exploit known inequalities to set safe boundaries.Consider:- The cosine function itself is bounded: \(-1 \leq \cos\left(\frac{1}{x}\right) \leq 1\) for any x.- Multiplying by \(x^2\) creates the inequality: \(-x^2 \leq x^2 \cos\left(\frac{1}{x}\right) \leq x^2\).These boundaries help dictate the oscillating range of f(x), boxing it within a neat perimeter to evaluate limits. This way, although the function might oscillate erratically near x = 0, these boundaries ensure it remains manageable and within predictable bounds.