Problem 17 - Calculus

Question

In Problems \(1-20\), find the numbers \(c\) that make \(f(x)\) into (A) a continuous function and (B) a differentiable function. In one case \(f(x) \rightarrow f(a)\) at every point, in the other case \(\Delta f / \Delta x\) has a limit at every point. $$ f(x)=\left\\{\begin{array}{cl} (c+\cos x) / x & x \neq 0 \\ 0 & x=0 \end{array}\right. $$

Step-by-Step Solution

Answer

\(c = 0\) makes the function both continuous and differentiable at \(x = 0\).

Step 1: Determine Continuity at x=0

For a function to be continuous at a point, three conditions must be satisfied: 1) The function must be defined at the point, 2) The limit of the function as it approaches the point must exist, and 3) The limit of the function as it approaches the point must equal the function's value at that point. For the given function, we need:1. \(f(0) = 0\) is given.2. Calculate \(\lim_{{x \to 0}}\frac{{c + \cos x}}{{x}}\). By using the limit property \(\lim_{{x \to 0}}\frac{{\cos x}}{{x}} = 0\) (since \(\cos 0 = 1\)), we get: \[ \lim_{{x \to 0}}\frac{{c + \cos x}}{{x}} = \lim_{{x \to 0}}\left(\frac{c}{x} + \frac{\cos x}{x}\right) \]3. For continuity, this limit must equal \(f(0) = 0\). Since the second part \(\frac{\cos x}{x}\) tends to infinity as \(x \to 0\), setting \(\frac{c}{x} = 0\) provides \(c = 0\).Therefore, \(c = 0\) is necessary for continuity.

Step 2: Check Differentiability at x=0

A function is differentiable at a point if it is continuous there and if the derivative exists at that point. The derivative exists if the limit of the difference quotient exists at that point.For the function to have a derivative at \(x = 0\), we need:\(\lim_{{h \to 0}}\frac{{f(h) - f(0)}}{h}\). Calculate:\[\lim_{{h \to 0}}\frac{{\frac{c + \cos h}{h} - 0}}{h} = \lim_{{h \to 0}}\frac{{c + \cos h}}{h^2}\]This limit must be finite. For this to happen, \(c = 0\) because the presence of \(\frac{c}{h^2}\) as \(h \to 0\) would yield infinity otherwise.Thus, \(c = 0\) also makes the function differentiable at \(x = 0\).

Key Concepts

Calculus

Calculus is one of the most important branches of mathematics that deals with change and motion. It provides the tools necessary to describe changes in quantities. At its core, calculus consists of two main concepts: differentiation and integration.
Differentiation focuses on the rate at which things change. For example, it helps in calculating velocities and accelerations. On the other hand, integration can be thought of as the accumulation of quantities, helping analyze areas under curves or the total quantity over time.
Calculus is widely used in various fields such as physics, engineering, economics, and biology, providing a framework to describe systems that are in motion or experiencing change.

Limit

The concept of a limit is fundamental to calculus. It describes the value that a function approaches as the input approaches a certain point. Limits help to define both continuity and differentiability.
For the function given in the exercise, to verify continuity at a point, we need to ensure that the limit of the function as it approaches the point equals the function's value at that point. In simpler terms, as we get closer and closer to a specific value on the x-axis, the limit tells us what value the y-output is getting closer to.
Limits are especially useful when dealing with indeterminate forms, like dividing by zero, and they form the basis of defining derivatives.

Differentiability

Differentiability is about how smoothly a function can be drawn on a graph. In simple terms, a function is differentiable at a point if it has a well-defined tangent line at that point. This means the function is smooth (no sharp corners or breaks) at that point.
For a function to be differentiable at a point, it must first be continuous there, and the slope between any two points (the difference quotient) must approach a definite value as the points get infinitely close.
In the given example, after verifying continuity at \(x = 0\), we further checked for differentiability, concluding that both required the element \(c\) to be zero to ensure a welldefined slope near \(x = 0\).

Continuous Function

A continuous function is a function that does not have any breaks, jumps, or holes. It's like drawing a curve on graph paper without lifting your pencil.
A function is continuous at a particular point if three things hold:

• The function is actually defined at that point.
• The function's limit exists as we approach that point.
• The limit of the function as we approach the point matches the actual value of the function at that point.

In our exercise, determining that \(c = 0\) allowed the function to meet these conditions at \(x = 0\), thus confirming its continuity there. Continuity is essential before considering differentiability at a point.