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Chapter 1: Problem 31

If \(f(x)=1 / x^{5},\) compute \(f(-2)\) and \(f^{\prime}(-2).\)

Short Answer

Expert verified
\ f(-2) = -\ \frac{1}{32} \ and \ f'(-2) = -\ \frac{5}{64}. \

Step by step solution

01

Calculate f(-2)

To find the value of the function at \( x = -2 \), substitute \( -2 \) into the function \( f(x) = \frac{1}{x^5} \). Calculate as follows: \ f(-2) = \frac{1}{(-2)^5} \.
02

Simplify f(-2)

Now simplify the expression \( f(-2) = \frac{1}{-32} = -\frac{1}{32} \). This gives us the value of the function at \( x = -2 \).
03

Compute the derivative \( f'(x) \)

Next, find the derivative of the function. The original function is \( f(x) = \frac{1}{x^5} = x^{-5} \). Using the power rule, \( f'(x) = -5x^{-6} \).
04

Calculate \ f'(-2) \

Finally, substitute \( x = -2 \) into the derivative \( f'(x) = -5x^{-6} \. \ f'(-2) = -5(-2)^{-6} = -5 \frac{1}{(-2)^6} = -5 \frac{1}{64} = -\frac{5}{64} \).

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Key Concepts

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

Power Rule for Differentiation
The power rule is a basic principle in calculus used to find the derivative of a function of the form \(f(x) = x^n\). If you want to differentiate this function, the power rule states that you bring down the exponent as a multiplier and then reduce the exponent by one. So, given \(f(x) = x^n\), the derivative \(f'(x)\) is \(nx^{n-1}\). For example, if \(f(x) = x^3\), then \(f'(x) = 3x^2\). This rule simplifies the process of differentiation and speeds up finding the rate of change.
Evaluating Functions
Evaluating a function involves finding the output value of a function given an input value for the variable. In other words, if you have a function \(f(x)\) and you want to find \(f(a)\) for some number \(a\), you substitute \(a\) into every instance of \(x\) in the function. For instance, with \(f(x) = x^2 + 3x + 1\), to find \(f(2)\), plug in \(2\) for \(x\): \[f(2) = 2^2 + 3(2) + 1 = 4 + 6 + 1 = 11\]
Negative Exponents
A negative exponent indicates that the base should be taken as the reciprocal raised to the opposite positive exponent. For example, \(x^{-n} = \frac{1}{x^n}\). This is useful in simplifying expressions and solving equations. In the context of differentiation, it helps to rewrite functions with negative exponents to utilize differentiation rules more easily. For instance, \(\frac{1}{x^5}\) can be written as \(x^{-5}\) before applying the power rule to find the derivative.
Basic Algebra
Basic algebra involves various fundamental concepts such as simplifying expressions, solving equations, and working with variables. Key operations include addition, subtraction, multiplication, and division with numbers or variables. For example, to simplify \(\frac{1}{(-2)^5}\), first compute \((-2)^5 = -32\), then find \(\frac{1}{-32} = -\frac{1}{32}\). Mastering these principles is essential for tackling more complex problems in calculus like differentiation and function evaluation.

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Most popular questions from this chapter

Let \(C(x)\) be the cost (in dollars) of manufacturing \(x\) items. Interpret the statements \(C(2000)=50,000\) and \(C^{\prime}(2000)=10 .\) Estimate the cost of manufacturing 1998 items.

Differentiate the function \(f(x)=\left(3 x^{2}+x-2\right)^{2}\) in two ways. (a) Use the general power rule. (b) Multiply \(3 x^{2}+x-2\) by itself and then differentiate the resulting polynomial.

If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x.\) $$f(x)=\frac{\sqrt{9+x}-\sqrt{9}}{x}, x \neq 0$$

Compute the following limits. $$\lim _{x \rightarrow \infty} \frac{1}{x-8}$$

For the given function, simultaneously graph the functions \(f(x), f^{\prime}(x),\) and \(f^{\prime \prime}(x)\) with the specified window setting. Note: since we have not yet learned how to differentiate the given function, you must use your graphing utility's differentiation command to define the derivatives. $$f(x)=\frac{x}{1+x^{2}},[-4,4] \text { by }[-2,2].$$
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