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Chapter 4: Problem 43

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

Short Answer

Expert verified
The expression simplifies and shows that \(\frac{f(x+h)-f(x)}{h} = 10^{x}\left(\frac{10^{h}-1}{h}\right).\)

Step by step solution

01

Identify the Function and Substitute

Given the function is \(f(x) = 10^x\). We want to find \(\frac{f(x+h) - f(x)}{h}\). Start by substituting these into the expression: \(f(x+h) = 10^{x+h}\) and \(f(x) = 10^x\).
02

Apply Function Substitutions

Substituting into the expression, we get: \[ \frac{f(x+h) - f(x)}{h} = \frac{10^{x+h} - 10^x}{h}. \]
03

Use Properties of Exponents

Recall the property of exponents that \(10^{x+h} = 10^x \cdot 10^h\). Replace \(10^{x+h}\) in the expression with \(10^x \cdot 10^h\).
04

Factor Out Common Terms

Factor out \(10^x\) from both terms in the numerator:\[ \frac{10^x \cdot 10^h - 10^x}{h} = 10^x \cdot \frac{10^h - 1}{h}.\]
05

Conclude the Algebra

The expression now matches the given right-hand side:\(10^x \left(\frac{10^h - 1}{h}\right).\)This concludes the derivation and shows the desired outcome.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

limits
Limits fundamentally help us understand the behavior of functions as they approach a specific point. In calculus, limits are essential in defining both derivatives and integrals.

When dealing with expressions like \( \frac{f(x+h) - f(x)}{h} \), which is called a difference quotient, we often want to understand what happens as \( h \) approaches zero. This limit process is vital in deriving the derivative of a function.

In practical terms, the limit checks the value that a function
  • approaches as the input approaches a certain value
  • helps in determining continuity, where if a function is continuous at a point, the limit at that point will equal the function's value
  • allows us to understand instantaneous rates of change, which are essential for finding the slope of a curve at a single point
Approaching zero in our example, you would compute the limit of the expression to verify how the derivative works, forming a core part of differentiating exponential functions.
derivatives
Derivatives reflect the rate of change of a function with respect to a variable. Think of derivatives as a tool to measure how a function behaves in an ever-changing environment, capturing its rate of change effectively.

For exponential functions like \( f(x) = 10^x \), the derivative gives us insight into how quickly the function grows or shrinks. The difference quotient \( \frac{f(x+h) - f(x)}{h} \) approaches the derivative as \( h \) approaches zero.

To clarify:
  • The basic concept involves recognizing the changes in the function value as the input changes.
  • Derivatives define tangent lines, which are the best linear approximations to the function at any given point.
  • They allow us to solve problems involving velocity and acceleration, or any scenario where change is involved.
In our equation from the original exercise, the solution effectively shows the derivative of the exponential function before plugging in \( h = 0 \) fully.
exponential functions
Exponential functions, characterized by the form \( a^x \), where \( a \) is a constant base, represent rapid growth or decay.

Given the function \( f(x) = 10^x \), it's a perfect example of exponential growth. Such functions frequently model real-world phenomena, including population growth, radioactive decay, and financial interest calculations.

Key aspects include:
  • Exponential functions continually multiply a constant base by itself, dictated by the power \( x \).
  • The derivative of an exponential function is constantly proportional to its own value, resulting in unique calculus properties.
  • The expression \( 10^{x+h} = 10^x \cdot 10^h \) uses exponentiation rules effectively, demonstrating property use to simplify calculus operations.
Ultimately, when performing calculus operations like differentiation, recognizing exponential function patterns helps streamline processes, as seen in transforming them into their derivative forms like in our previous solution.

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Most popular questions from this chapter

These exercises deal with logarithmic scales. The noise from a power mower was measured at 106 dB. The noise level at a rock concert was measured at 120 dB. Find the ratio of the intensity of the rock music to that of the power mower.

A sum of \(\$ 5000\) is invested at an interest rate of \(9 \%\) per year, compounded semiannually. (a) Find the value \(A(t)\) of the investment after \(t\) years. (b) Draw a graph of \(A(t)\) (c) Use the graph of \(A(t)\) to determine when this investment will amount to \(\$ 25,000\).

Take logarithms to show that the equation $$x^{1 / \log x}=5$$ has no solution. For what values of \(k\) does the equation $$x^{1 / \log x}=k$$ have a solution? What does this tell us about the graph of the function \(f(x)=x^{1 / \log x ?}\) Confirm your answer using a graphing device.

A small lake is stocked with a certain species of fish. The fish population is modeled by the function $$P=\frac{10}{1+4 e^{-0.8 t}}$$ where \(P\) is the number of fish in thousands and \(t\) is measured in years since the lake was stocked. (a) Find the fish population after 3 years. (b) After how many years will the fish population reach 5000 fish?

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$\log _{2} 5$$
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