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Chapter 5: Problem 13

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=\frac{x+\pi}{2} $$

Short Answer

Expert verified
The derivative, \(f'(x)\), equals to \(\frac{1}{2}\). The derivative at \(x = 0\), \(f'(0)\), is \(\frac{1}{2}\). The derivative at \(x = 2\), \(f'(2)\), is \(\frac{1}{2}\). The derivative at \(x = -1\), \(f'(-1)\), is \(\frac{1}{2}\).

Step by step solution

01

Find the derivative of the function

The derivative of the function \(f(x) = \frac{x+\pi}{2}\) can be obtained using the power rule. The power rule states that the derivative of \(x^n\) is \(n \cdot x^{n-1}\). In this case, \(n = 1\), so the derivative of \(x\) is \(1\). The derivative of a constant is \(0\), so the derivative of \(\pi\) is \(0\). The result is the derivative, \(f'(x)\), equal to \(\frac{1}{2}\).
02

Find the derivative at \(x = 0\)

To find \(f'(0)\), substitute \(x = 0\) in the derivative function. So, \(f'(0)\) equals to \(\frac{1}{2}\).
03

Find the derivative at \(x = 2\)

To find \(f'(2)\), substitute \(x = 2\) in the derivative function. So, \(f'(2)\) equals to \(\frac{1}{2}\).
04

Find the derivative at \(x = -1\)

To find \(f'(-1)\), substitute \(x = -1\) in the derivative function. So, \(f'(-1)\) equals to \(\frac{1}{2}\).

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Key Concepts

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

Understanding Derivatives
A derivative represents the rate at which a function changes at any given point. Think of it as a snapshot of how quickly something is happening. By finding the derivative of a function, we can understand how the output of the function responds to changes in its input.
For example, if we know the derivative of a function about speed, we can find out how quickly the speed changes, which is effectively acceleration. When dealing with a straightforward linear function like the one in the exercise, the derivative indicates a consistent rate of change.
  • It helps in understanding and analyzing the behavior of functions.
  • Calculating the derivative is essential in different fields, from physics to economics.
The Power Rule: A Handy Tool
The power rule is a quick method for finding the derivative of functions that involve exponents. It simplifies the process, especially for polynomial terms. The rule states: if you have a term in the form of \(x^n\), the derivative is \(n \cdot x^{n-1}\).
In the example function \(f(x) = \frac{x+\pi}{2}\), we identify that \(x\) is raised to the power of one. By applying the power rule, the derivative of \(x\) is simply \(1\) since \(1 \cdot x^{1-1} = 1\).
  • This makes handling terms with powers very straightforward.
  • It's a fundamental rule used extensively in calculus.
The power rule doesn't apply to constants, which brings us to our next topic.
Derivatives of Constant Functions
In calculus, a constant function is a function that does not change, regardless of the input. The derivative of a constant is always zero because a constant doesn鈥檛 change; it has no rate of change.
For example, the number \(\pi\) is a constant in the function \(f(x) = \frac{x+\pi}{2}\). When taking its derivative, it results in zero, since constants don't vary and have no slope.
  • Every constant's derivative is zero, reflecting no change.
  • It's important to remember constants disappear when taking derivatives.
Understanding this helps in computing derivatives efficiently and avoiding mistakes.

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Most popular questions from this chapter

Let \(f(x)=\frac{1}{x} .\) In this problem we will look at the slope of the tangent line to \(f(x)\) at point \(P=\left(\frac{1}{2}, 2\right)\) (a) Is the slope of the tangent line to \(f\) at \(P\) positive, or negative? (b) By calculating the slope of the secant line through \(P\) and a nearby point on the graph of \(f\), approximate \(f^{\prime}\left(\frac{1}{2}\right)\). First choose the point with an \(x\) -coordinate of \(0.49 .\) Next choose the point with an \(x\) -coordinate of \(0.501 .\) Now produce an approximation that is better than either of the previous two. (c) By calculating the limit of the difference quotient, nd \(f^{\prime}\left(\frac{1}{2}\right)\). (d) Find the equation of the tangent line to \(f\) at \(P\).

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=x^{2} $$

In Problems 5 through 8 , estimate \(f^{\prime}(c)\) by calculating the difference quotient \(\frac{f(c+h)-f(c)}{h}\) for successively smaller values of \(|h| .\) Use both positive and negative values of \(h .\). $$ f(x)=\sqrt[3]{x} \text { . Approximate } f^{\prime}(8) $$

Use the limit de nition of derivative to show that the derivative of the linear function \(f(x)=a x+b\) is \(a\). Why is this exactly what you would expect? You have shown that the derivative of a constant is zero. Explain, and explain why this is exactly what you would expect.

You have formed a study group with a few of your friends. One of the people in your study group has been ill for the past \(2 \frac{1}{2}\) weeks and is concerned about the upcoming examination. She needs to understand the main ideas of the past few weeks. Your essay should be designed to help her. (a) Explain the relationship between average rate of change and instantaneous rate of change, and between secant lines and tangent lines. Your classmate is unclear why the definition of derivative involves some limit with \(h\) going to zero. She wants to know why you can't just set \(h=0\) to begin with and be done with it. Why does calculus involve this limit business? (b) She also has one specific question. She is not clear about how you get the graph of \(f^{\prime}\) from the graph of \(f\). Just before she got sick we were doing that. She believes that if the graph of \(f^{\prime}\) is increasing, then the graph of \(f\) is also increasing. Her Course Assistant says this is wrong, but she claims that sometimes she gets the right answer using this reasoning. What's the story? How to write this essay: First, think about your friend's position. Particularly in part (a) see if you can understand what is confusing her and how to clarify it for her. Outline your answer. Then write your essay. Use words precisely. Try to avoid pronouns. For example, do not say "it" is increasing; be specific \(-\) what is increasing? Use words to say precisely what you mean. The real purpose of this essay: These are issues that are important for you to understand. We want you to put together what you have learned in your own words. We also want you to learn to write about mathematics by using words precisely to say exactly what you mean.
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