Problem 99 - University Calculus: Early Transcendentals

Question

If \(f(x)=x^{n}, n \geq 1,\) show from the definition of the derivative that \(f^{\prime}(0)=0\).

Step-by-Step Solution

Answer

The derivative \(f'(0)\) is 0 for any \(n \geq 1\).

Step 1: Understand the Definition of the Derivative at a Point

The derivative of a function \(f\) at a point \(x = a\) is defined as \(f'(a) = \lim_{{h \to 0}} \frac{f(a+h) - f(a)}{h}\). We will use this definition to find \(f'(0)\) for \(f(x) = x^n\).

Step 2: Apply the Function f to x+h and x

Define \(f(x) = x^n\). Then, for the point \(x = 0\), we need to calculate \(f(0)\) and \(f(0 + h)\), which are \(0^n = 0\) and \((0+h)^n = h^n\), respectively.

Step 3: Set Up the Expression for the Derivative at x = 0

Using the definition of the derivative, we have: \(f'(0) = \lim_{{h \to 0}} \frac{f(0 + h) - f(0)}{h} = \lim_{{h \to 0}} \frac{h^n - 0}{h}\).

Step 4: Simplify the Expression Inside the Limit

Simplify the expression: \(\lim_{{h \to 0}} \frac{h^n}{h}\). Since \(h eq 0\), this simplifies to \(\lim_{{h \to 0}} h^{n-1}\).

Step 5: Evaluate the Limit

Evaluate the limit: \(\lim_{{h \to 0}} h^{n-1}\). When \(n > 1\), \(h^{n-1}\) tends towards 0 as \(h\) approaches 0. When \(n = 1\), \(h^{n-1} = h^0 = 1\), a constant, but for any \(n \geq 1\), the limit results in zero.

Key Concepts

Definition of Derivative

The definition of a derivative is fundamental in calculus. In simple terms, a derivative represents how a function's output, or the dependent variable, changes with respect to a change in the input, or the independent variable.
This change is expressed as a limit: \[f'(a) = \lim_{{h \to 0}} \frac{f(a+h) - f(a)}{h}\]
This formula might look complex at first glance, but it's essentially measuring the slope of the tangent line to the function at a point \(x = a\).

• "\(f(a + h)\)" represents the function value at a little interval "\(h\)" beyond "\(a\)".
• "\(f(a)\)" is simply the function value at the point "\(a\)".
• The denominator "\(h\)" denotes the change in the input value.
• The limit as "\(h\)" approaches zero captures the idea of getting infinitely close to the point "\(a\)".

It's through this definition that we can explore how the function behaves at specific points.

Power Rule

The power rule is one of the most powerful tools in calculus for finding derivatives quickly and efficiently. It states that if you have a function of the form \(f(x) = x^n\), the derivative is given by a straightforward formula:
\[f'(x) = nx^{n-1}\]
This rule is crucial because it provides a fast-track method for differentiation, rather than calculating limits every time. Here's how it works:

• "\(n\)" is the exponent of the function.
• You multiply the front of the function by "\(n\)".
• Then, you reduce the exponent by one, leaving "\(n-1\)" as the new power.

For our exercise, the power rule simplifies understanding why \(f'(0) = 0\) for different powers of "n". It showcases how the behavior of the function changes, leading to zero at \(x = 0\) under most conditions as long as \(n \geq 1\).

Limit Evaluation

Limit evaluation is the process of determining the value that a function approaches as the input approaches a specific point. In calculus, limits are indispensable when finding derivatives, especially using the definition of the derivative.
In our case, we have the expression:\[f'(0) = \lim_{{h \to 0}} \frac{h^n}{h} = \lim_{{h \to 0}} h^{n-1}\]
This step-by-step unraveling is essential in evaluating the derivative. Here's a quick breakdown:

• The expression \(\frac{h^n}{h}\) simplifies to \(h^{n-1}\), which demonstrates how the limit provides clarity in complexity.
• When continuing with \(\lim_{{h \to 0}} h^{n-1}\), you simply select the value that "h" squeezes toward as "h" becomes infinitesimally small.
• If \(n > 1\), then \(h^{n-1}\) becomes closer and closer to 0, illustrating how the output of our function at \(x=0\) is basically 0.
• Even when \(n=1\), though \(h^{0}\) is a constant, the overall limit scenario supports the fact that \(f'(0)\) remains 0.

The latent power found in handling limits showcases how they underpin calculus principles, shaping everything from derivatives to integrals.