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Chapter 1: Problem 10

Show how to evaluate the function $$ f(x)=2 e^{4 x}-e^{3 x}+5 e^{x}+1 $$ efficiently. Hint: Consider letting \(z=e^{x}\).

Short Answer

Expert verified
Use \( z = e^x \) to simplify and evaluate.

Step by step solution

01

- Substitute with a new variable

Let鈥檚 start by substituting the variable. Let \( z = e^{x} \). This changes the original function \( f(x) \) to a function of \( z \).
02

- Rewrite the function

Using the substitution \( z = e^{x} \), rewrite the function. The function becomes: \[ f(z) = 2z^4 - z^3 + 5z + 1 \]
03

- Evaluate the new function

To find \( f(x) \), we'll now evaluate the simplified function \( f(z) \). Replace \( z \) back with \( e^x \) after evaluating the polynomial, if necessary.

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Key Concepts

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

Variable Substitution
Variable substitution is a powerful tool in numerical analysis for simplifying complex functions. Instead of dealing with the original variables, we introduce a new variable to make calculations easier. In this exercise, we use the substitution:

Let \( z = e^{x} \).

Using this new variable, we transform the original function \( f(x) \) into a simpler function of \( z \). This makes it much easier to handle, as we see in the subsequent steps. This approach is especially useful when dealing with exponential functions, as it transforms them into polynomial functions that are often simpler to manage.
Polynomial Transformation
Polynomial transformation leverages substitution to turn an exponential function into a polynomial one. Once we've substituted \( z = e^{x} \) into the original function, we rewrite it as:

\( f(z) = 2z^4 - z^3 + 5z + 1 \).

Polynomials are generally easier to evaluate and manipulate compared to exponential functions. This is because polynomial arithmetic is straightforward, involving basic operations such as addition, subtraction, and multiplication. By converting the function into a polynomial, we take advantage of numerous algebraic techniques that make the function evaluation process more efficient and less error-prone.
Exponential Functions
Exponential functions, like \( e^{x} \), grow very quickly and are fundamental in many areas of mathematics and science. In the original function:

\( f(x) = 2 e^{4 x} - e^{3 x} + 5 e^{x} + 1 \),

the exponential terms \( e^{4x}, e^{3x} \), and \( e^{x} \) can be cumbersome to work with directly. By employing variable substitution and transforming the function into a polynomial, we sidestep the challenges of dealing with rapidly growing exponential terms. After evaluating the polynomial form, we can always substitute back to get the original variable if needed. This method significantly simplifies the problem, making it more approachable for students.

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Most popular questions from this chapter

(a) Obtain a Taylor polynomial with remainder for \(f(t)=1 /\left(1+t^{2}\right)\), about \(a=0\). Hint: Substitute \(x=-t^{2}\) into \((1.16)\). (b) Obtain a Taylor polynomial with remainder for \(g(x)=\tan ^{-1} x .\) Do this by integrating the result in (a) and using $$ \tan ^{-1}(x)=\int_{0}^{x} \frac{d t}{1+t^{2}} $$

Rewrite \(f(x)=1+x^{6}\) in the form $$ f(x)=\sum_{j=0}^{6} a_{j}(x-1)^{j} $$ Hint: Expand \(f(x)\) as a Taylor polynomial of degree 6 about \(x=1\). Using this, what are the coefficients \(\left\\{a_{j}\right\\}\) ? What is the error \(f(x)-p_{6}(x)\) in this case?

The quotient $$ g(x)=\frac{\log (1+x)}{x} $$ is undefined for \(x=0 .\) Approximate \(\log (1+x)\) using Taylor polynomials of degrees 1,2, and 3 , in turn, to determine a natural definition of \(g(0)\).

(a) Let the polynomial \(p(x)\) be an even function, meaning that \(p(-x)=p(x)\) for all \(x\) of interest. Show this implies that the coefficients are zero for all terms of odd degree. (b) Let the polynomial \(p(x)\) be an odd function, meaning that \(p(-x)=-p(x)\) for all \(x\) of interest. Show this implies that the coefficients are zero for all terms of even degree. (c) Let \(p(x)=a_{0}+a_{1} x+a_{2} x^{2}+a_{5} x^{5} .\) Give conditions on the coefficients \(\left\\{a_{0}, a_{1}, a_{2}, a_{5}\right\\}\) so that \(p(x)\) is even. Repeat with \(p(x)\) being odd.

Use Taylor polynomials with remainder term to evaluate the following limits: (a) \(\lim _{x \rightarrow 0} \frac{1-\cos (x)}{x^{2}}\) (b) \(\lim _{x \rightarrow 0} \frac{\log \left(1+x^{2}\right)}{2 x}\) (c) \(\lim _{x \rightarrow 0} \frac{\log (1-x)+x e^{x / 2}}{x^{3}}\) Hint: Use Taylor polynomials for the standard functions [e.g., \(\cos (t), \log (1+t)\), and \(\left.e^{t}\right]\) to obtain polynomial approximations to the numerators of these fractions; and then simplify the results
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