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Chapter 14: Problem 56

Find \(\frac{f(x+h)-f(x)}{h}\) and simplify. $$f(x)=x^{5}+8$$

Short Answer

Expert verified
The simplified form of \(\frac{f(x+h) - f(x)}{h}\) for the function \(f(x) = x^{5} + 8\) is \(5x^{4}\)

Step by step solution

01

Find \(f(x+h)\)

Firstly, the value of \(f(x+h)\) is computed by substituting \(x+h\) in place of x in the original function. It is done as follows: \(f(x+h) = (x+h)^{5} + 8\)
02

Expand \((x + h)^{5}\)

Next, expand \((x + h)^{5}\) using the binomial theorem, which states that \((x + h)^{5} = x^{5} + 5x^{4}h + 10x^{3}h^{2} + 10x^{2}h^{3} + 5xh^{4} + h^{5}\). So, \(f(x+h) = x^{5} + 5x^{4}h + 10x^{3}h^{2} + 10x^{2}h^{3} + 5xh^{4} + h^{5} + 8\)
03

Subtract \(f(x)\)

Then, \(f(x)\) is subtracted from \(f(x+h)\) which gives: \(f(x+h) - f(x) = [x^{5} + 5x^{4}h + 10x^{3}h^{2} + 10x^{2}h^{3} + 5xh^{4} + h^{5} + 8] - [x^{5} + 8] = 5x^{4}h + 10x^{3}h^{2} + 10x^{2}h^{3} + 5xh^{4} + h^{5}\)
04

Divide by h

The next step is to find the difference quotient by dividing \(f(x+h) - f(x)\) by h. By factoring out h, this simplifies into: \( \frac{5x^{4}h + 10x^{3}h^{2} + 10x^{2}h^{3} + 5xh^{4} + h^{5}}{h} = 5x^{4} + 10x^{3}h + 10x^{2}h^{2} + 5xh^{3} + h^{4}\)
05

Simplify as h approaches 0

As the limit of the difference quotient as h approaches zero is the derivative, all terms involving h will become zero. Therefore, the simplified answer is: \(5x^{4}\)

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Most popular questions from this chapter

Use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of \(f\) and discuss its relationship to the sum of the given series. Function $$f(x)=\frac{4\left[1-(0.6)^{x}\right]}{1-0.6}$$ Series $$\begin{array}{l}4+4(0.6)+4(0.6)^{2} \\\\+4(0.6)^{3}+\cdots\end{array}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I was able to find the sum of the first fifty terms of an arithmetic sequence even though I did not identify every term.

Use the formula for the general term (the nth term) of a geometric sequence to solve. You are offered a job that pays \(\$ 30,000\) for the first year with an annual increase of \(5 \%\) per year beginning in the second year. That is, beginning in year \(2,\) your salary will be 1.05 times what it was in the previous year. What can you expect to earn in your sixth year on the job? Round to the nearest dollar.

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$5,7,9,11, \dots$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the \(n\) th term of a geometric sequence is \(a_{n}=3(0.5)^{n-1}\) the common ratio is \(\frac{1}{2}\).
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