Problem 40 - Precalculus Mathematics for Calculus

Question

$$\text { If } f(x)=10^{x}, \text { show that } \frac{f(x+h)-f(x)}{h}=10^{x}\left(\frac{10^{h}-1}{h}\right)$$

Step-by-Step Solution

Answer

The expression is verified as \( 10^x \left( \frac{10^h - 1}{h} \right) \).

Step 1: Start with the difference quotient

The difference quotient is defined as \( \frac{f(x+h) - f(x)}{h} \). For the function \( f(x) = 10^x \), substitute \( f(x+h) = 10^{x+h} \) and \( f(x) = 10^x \). This yields: \( \frac{10^{x+h} - 10^x}{h} \).

Step 2: Simplify the expression

The expression \( \frac{10^{x+h} - 10^x}{h} \) can be rewritten using the property of exponents: \( 10^{x+h} = 10^x \cdot 10^h \). Substituting this gives \( \frac{10^x \cdot 10^h - 10^x}{h} \).

Step 3: Factor out the common term

Notice that \( 10^x \) is a common factor in the numerator. Factor it out to get: \( \frac{10^x (10^h - 1)}{h} \).

Step 4: Rewrite the expression

Now, the expression is \( 10^x \cdot \frac{10^h - 1}{h} \). This matches the form \( 10^x \left( \frac{10^h - 1}{h} \right) \) we needed to prove. Thus, the expression is verified.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where variables appear as exponents. These functions are powerful tools for modeling growth processes, like population growth or compound interest. The general form is given by \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the variable exponent.
For example, in the function \( f(x) = 10^x \), 10 is the base and \( x \) is the exponent. This function represents exponential growth because as \( x \) increases, \( 10^x \) increases exponentially. Exponential functions can rapidly increase, which makes them useful for representing dynamic situations in many scientific and mathematical fields.
Understanding exponential functions requires familiarity with exponential growth behavior. As \( x \) increases in small increments, \( 10^x \) grows much faster than linear or even quadratic functions, which is a unique characteristic of exponential functions.

Properties of Exponents

Exponents have properties that can simplify mathematical expressions and equations. Understanding these properties is crucial when working with exponential functions like \( 10^x \). Here are the key properties:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \). Multiplying numbers with the same base adds their exponents.
• Power of a Power: \( (a^m)^n = a^{m \, n} \). Raising a power to another power multiplies the exponents.
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \). Dividing numbers with the same base subtracts the exponents.
• Power of a Product: \( (ab)^n = a^n \cdot b^n \). The exponent applies to both factors in a product.
• Zero Exponent: \( a^0 = 1 \). Any non-zero number raised to the power of zero is one.

In our exercise, one of these properties, the product of powers, simplifies the expression by writing \( 10^{x+h} \) as \( 10^x \cdot 10^h \). This understanding is essential for simplifying complex expressions in exponential functions.

Factoring

Factoring is a mathematical technique used to simplify expressions by finding common factors in terms. In the context of the difference quotient expression, factoring can simplify the problem considerably.
Consider the expression \( 10^x \cdot 10^h - 10^x \). Here, \( 10^x \) is a common factor in both terms. By factoring it out, the problem becomes:

• \( 10^x (10^h - 1) \)

This process of factoring is essential in breaking down complex expressions into simpler, more manageable parts. It is particularly useful in algebraic expressions and equations, making further simplification or solving equations possible.
Students often encounter factoring in polynomial equations, where identifying common factors can drastically reduce the problem's complexity.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form while keeping their fundamental properties intact. This process is critical in mathematics for making expressions easier to work with and understand.
To simplify an expression properly:

• Identify patterns or common terms in all parts of the expression.
• Use algebraic properties, like the properties of exponents or factoring techniques, to reduce complexity.
• Ensure the expression remains equivalent to the original.

In our exercise, the simplification process was shown by refactoring the expression \( \frac{10^x (10^h - 1)}{h} \) into \( 10^x \cdot \frac{10^h - 1}{h} \). This translation illustrates the essence of simplification: making expressions neat and tidy without altering their mathematical essence.
Simplifying expressions helps students recognize underlying patterns and relationships within mathematical structures and is a vital skill in solving algebraic problems efficiently.