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Chapter 4: Problem 84

Exer. 83-84: Approximate the function at the value of \(x\) to four decimal places. (a) \(f(x)=\log \left(2 x^{2}+1\right)-10^{-x}, \quad x=1.95\) (b) \(g(x)=\frac{x-3.4}{\ln x+4}, \quad x=0.55\)

Short Answer

Expert verified
(a) \( f(1.95) \approx 1.3574 \); (b) \( g(0.55) \approx -0.8428 \).

Step by step solution

01

Substitute into the function for Part (a)

We begin by substituting the given value of \( x = 1.95 \) into the function \( f(x) = \log(2x^2 + 1) - 10^{-x} \). This will allow us to approximate the function's value at this specific \( x \).
02

Calculate the components of f(x) for Part (a)

First, compute \( 2 \cdot (1.95)^2 + 1 \). Then, use the logarithmic function on this result. Finally, evaluate \( 10^{-1.95} \) and subtract this value from the logarithmic result to get \( f(1.95) \).
03

Substitute into the function for Part (b)

Next, substitute \( x = 0.55 \) into the function \( g(x) = \frac{x - 3.4}{\ln x + 4} \). This setups the expression we need to compute for \( g(0.55) \).
04

Calculate the components of g(x) for Part (b)

First, find \( 0.55 - 3.4 \). Then, calculate \( \ln 0.55 + 4 \). Use these results to find \( g(0.55) = \frac{0.55 - 3.4}{\ln 0.55 + 4} \).
05

Approximate and round the results

Finish by performing the calculations from the previous steps and round the results of both \( f(1.95) \) and \( g(0.55) \) to four decimal places.

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Key Concepts

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

Logarithmic Function
Logarithmic functions are a key component in mathematics, especially when it comes to modeling and interpreting real-world phenomena. A logarithmic function is the inverse of an exponential function. For a function like \( f(x) = \log (2x^2 + 1) - 10^{-x} \), the notation \( \log \) typically refers to a logarithm with a specific base. If no base is given, it might imply either a common logarithm (base 10) or a natural logarithm (base \( e \)).
Hence, it鈥檚 important to know the context or consult instructions if it's within a specific setting, like your textbook.
  • Exponential growth tends to happen quickly while logarithmic growth happens slowly.
  • When dealing with log functions, the operations inside the argument, such as \( 2x^2 + 1 \), need to be computed before applying the log itself.
Understanding the nature of logarithms will aid in calculating values and approximating functions accurately.
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is another critical concept used across various mathematical and scientific domains. This logarithm has a base \( e \) (approximately 2.71828), and is used when dealing with certain growth patterns, decay processes, and in this exercise, function approximation.
Calculating \( \ln(x) \) means finding the power to which \( e \) must be raised to get \( x \). For instance, in the given function \( g(x) = \frac{x-3.4}{\ln x + 4} \), evaluating \( \ln(0.55) \) involves this natural logarithm.
  • The value of \( e \) itself is a transcendental number and appears naturally in scenarios involving continuous growth or compounding.
  • Using a scientific calculator or software is crucial when obtaining precise \( \ln \) values needed for exact calculations.
Rounding Numbers
Rounding is an essential mathematical technique used to approximate numbers by decreasing their number of decimal places, making them simpler or more convenient for a given context. In the context of this exercise, results are to be rounded to four decimal places for both functions.
When rounding, one typically looks at the fifth decimal place to decide whether to round the fourth place up or leave it as is.
  • If the digit in the next decimal place (fifth place) is 5 or higher, you round up.
  • Otherwise, you leave the current digit as it is.
Practicing this can help ensure that your approximated function values are close enough to be accurate and useful for further applications.
Evaluation of Expressions
Evaluating expressions involves substituting values into the given mathematical function and calculating the result. This task can help understand the behavior of functions under specific circumstances.
In this task, we have two functions, \( f(x) = \log(2x^2 + 1) - 10^{-x} \) and \( g(x) = \frac{x-3.4}{\ln x + 4} \), to be evaluated at given \( x \)-values.
  • Start by substituting the given \( x \) values into their respective functions.
  • Calculate any operations inside the parentheses first, and follow the order of operations.
  • Keep track of negative signs and division carefully, as mistakes in these areas often lead to incorrect results.
Taking each step methodically helps you arrive at correct results and better understand the working mechanics of the expression.

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Most popular questions from this chapter

Let \(n\) be any positive integer. Find the inverse function of \(f\) if (a) \(f(x)=x^{n}\) for \(x \geq 0\) (b) \(f(x)=x^{m / n}\) for \(x \geq 0\) and \(m\) any positive integer

Some lending institutions calculate the monthly payment \(M\) on a loan of \(L\) dollars at an interest rate \(r\) (expressed as a decimal) by using the formula $$M=\frac{L r k}{12(k-1)}$$ where \(k=[1+(r / 12)]^{12 t}\) and \(t\) is the number of years that the loan is in effect. (a) Find the monthly payment on a 30 -year 250,000 dollars home mortgage if the interest rate is \(8 \%\) (b) Find the total interest paid on the loan in part (a).

Urban population density An urban density model is a formula that relates the population density \(D\) (in thousands/mi \(^{2}\) ) to the distance \(x\) (in miles) from the center of the city. The formula \(D=a e^{-b x}\) for central density \(a\) and coefficient of decay \(b\) has been found to be appropriate for many large U.S. cities. For the city of Atlanta in 1970 , \(a=5.5\) and \(b=0.10 .\) At approximately what distance was the population density 2000 per square mile?

Radio stations The table lists the total numbers of radio stations in the United States for certain years. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number } \\\\\hline 1950 & 2773 \\\\\hline 1960 & 4133 \\\\\hline 1970 & 6760 \\\\\hline 1980 & 8566 \\\\\hline 1990 & 10,770 \\\\\hline 2000 & 12,717 \\\\\hline\end{array}$$ (a) Plot the data. (b) Determine a linear function \(f(x)=a x+b\) that models these data, where \(x\) is the year. Plot \(f\) and the data on the same coordinate axes. (c) Find \(f^{-1}(x) .\) Explain the significance of \(f^{-1}\) (d) Use \(f^{-1}\) to predict the year in which there were \(11,987\) radio stations. Compare it with the true value, which is 1995

Exer. \(85-86:\) Approximate the real root of the equation. $$x \ln x=1$$
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