Problem 37 - Calculus: Early Transcendentals

Question

Each limit represents the derivative of some function \( f \) at some number \( a \). State such an \( f \) and \( a \) in each case. \( \displaystyle \lim_{h \to 0} \frac{\sqrt{9 + h} - 3}{h} \)

Step-by-Step Solution

Answer

Function is \( f(x) = \sqrt{x} \), evaluated at \( a = 9 \).

Step 1: Identify the Derivative Form

The expression \( \lim_{h \to 0} \frac{\sqrt{9 + h} - 3}{h} \) matches the definition of a derivative \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \). This is used to find \( f'(a) \).

Step 2: Determine Function \( f(x) \)

To find the function \( f \), notice that \( f(a+h) = \sqrt{9 + h} \). Comparing this with \( f(x) = \sqrt{x} \), we can see that \( a + h = x \) implies \( x = 9 + h \). Thus, \( f(x) = \sqrt{x} \).

Step 3: Determine the Point \( a \)

In \( f(a) \), the expression \( \sqrt{9} \) is used, which implies that \( a = 9 \) since the derivative is evaluated at \( x = 9 \).

Step 4: Match Expression to Derivative at Point

The expression \( \lim_{h \to 0} \frac{f(9+h) - f(9)}{h} \) is recognized as the derivative \( f'(9) \) for \( f(x) = \sqrt{x} \). Therefore, \( f'(a) = f'(9) \).

Key Concepts

Limit Definition of Derivative

The limit definition of the derivative is fundamental in calculus. It provides a way to compute the derivative of a function at a specific point. This definition is expressed as:\[\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\]Here, \( f(a+h) \) and \( f(a) \) represent the function evaluated at slightly different points. The concept revolves around observing how small changes in the input \( h \) affect the output. This gives the rate of change of the function at the point \( a \). It provides a bridge between the finite changes and an instantaneous rate of change, a crucial idea in calculus.

In practical terms:

• "\( h \to 0 \)" means we consider infinitesimally small values of \( h \).
• The expression \( \frac{f(a+h) - f(a)}{h} \) represents the slope between two close points on the function.
• Taking the limit as \( h \to 0 \) gives us the exact slope at point \( a \).

By understanding this, you can find derivatives, which describe how "fast" or "slow" things are changing.

Square Root Function

The square root function is denoted as \( f(x) = \sqrt{x} \). It is a fundamental mathematical function with a unique property: for a given number, it outputs the value which, when multiplied by itself, gives the original number.

Key properties include:

• It is only defined for non-negative values (\( x \geq 0 \)) because the square root of negative numbers is not real.
• It has a gentle curve, starting from \( (0, 0) \), that rises to the right.
• Its derivative becomes smaller as \( x \) increases, meaning the function flattens out as you move to the right.

The square root function is common in real-world applications where quantities grow rapidly initially and then slow down, like time dilation or certain acceleration models. This function's rate of change is significant when understanding calculus problems involving curves.

Evaluating Limits

Evaluating limits is a cornerstone of calculus and helps in understanding how functions behave as inputs approach certain values. When we evaluate limits, we determine the value that a function approaches as the independent variable gets infinitely close to a given point.

In the context of our problem:

• We looked at \( \lim_{h \to 0} \frac{\sqrt{9 + h} - 3}{h} \), which is structured to follow the limit definition of the derivative.
• This form involves direct substitution, algebraic manipulation, or rationalization to find the limit.
• For this problem, we need\( h \to 0 \), making the fraction approach the value of the derivative at that point.

Successful evaluation of limits is essential in finding derivatives, integrals, and solving differential equations. It captures how a function behaves at transitional points, often those involving indeterminate forms.

Differentiation

Differentiation is a fundamental operation in calculus, enabling us to compute the derivative of a function. The derivative represents the rate of change or slope of the function at any given point, which is crucial for understanding dynamic systems.

For the square root function \( f(x) = \sqrt{x} \):

• The derivative, obtained via differentiation, describes how quickly \( \sqrt{x} \) changes with respect to \( x \).
• Using the limit definition, we find its derivative to be \( f'(x) = \frac{1}{2\sqrt{x}} \).
• At \( x = 9 \), the derivative gives \( f'(9) = \frac{1}{2 \times \sqrt{9}} = \frac{1}{6} \), indicating that as \( x \) approaches 9, the function changes at the rate of one-sixth.

Differentiation is a cornerstone technique in calculus, essential for analyzing and modelling how variables interact and change together in science and engineering fields.