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Chapter 1: Problem 27

If \(f(x)=x^{3}\), compute \(f(-5)\) and \(f^{\prime}(-5)\).

Short Answer

Expert verified
The value of \( f(-5) \) is -125, and the derivative \( f'(-5) \) is 75.

Step by step solution

01

Understanding the Function

The function given is a cubic function, defined by the equation \( f(x) = x^3 \).
02

Compute \( f(-5) \)

To find \( f(-5) \), substitute \( x = -5 \) into the function. \[ f(-5) = (-5)^3 \] Simplify the expression: \[ f(-5) = -125 \]
03

Find the Derivative \( f'(x) \)

To find \( f'(-5) \), first find the derivative of the function. The derivative of \( f(x) = x^3 \) can be found using the power rule, which states \( \frac{d}{dx}[x^n] = nx^{n-1} \). Applying the power rule: \[ f'(x) = 3x^2 \]
04

Compute \( f'(-5) \)

To find \( f'(-5) \), substitute \( x = -5 \) into the derivative. \[ f'(-5) = 3(-5)^2 \] Simplify the expression: \[ f'(-5) = 3 \times 25 = 75 \]

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Key Concepts

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

Cubic Function
A cubic function is a type of polynomial, specifically a polynomial of degree three. It can be expressed in the form \( f(x) = ax^3 + bx^2 + cx + d \), where a, b, c, and d are constants, and a is not zero. In our specific case, the function given is \( f(x) = x^3 \).

Cubic functions typically have one real root and two complex roots, or three real roots. They are known for their characteristic 'S' shape when graphed, which shows that they increase at one end and decrease at the other.

Key characteristics of cubic functions:
  • They have the highest degree of three.
  • The leading coefficient determines the end behavior of the graph.
  • They can have up to three turning points and two inflections.
Understanding these basic properties helps when analyzing or differentiating cubic functions.
Power Rule
The power rule is a fundamental concept in calculus used to find the derivative of polynomial functions. It simplifies the process of differentiation significantly. The power rule states: \[ \frac{d}{dx}[x^n] = nx^{n-1} \] where n is a constant real number.

This rule is especially useful with functions of the form \(x^n\) where n is a positive integer, like our cubic function \(f(x) = x^3\).
To differentiate our function using the power rule:
  • Given \( f(x) = x^3 \), the exponent n is 3.
  • The derivative, according to the power rule, is \[ f'(x) = 3x^{3-1} = 3x^2 \]
This simplified approach saves time and ensures accuracy when computing derivatives.
Derivative Computation
Derivative computation involves calculating the derivative of a function to understand how it changes at any given point. For the cubic function \( f(x) = x^3 \), we computed its derivative using the power rule to get \( f'(x) = 3x^2 \).

Once we have the derivatives, we can evaluate them at specific points. For example, to find the derivative at \(x = -5\), we substitute \(-5\) into the derivative: \[ f'(-5) = 3(-5)^2 \]

Hence, we get:
  • \( f'(-5) = 3 \times 25 = 75 \)

This tells us that the rate of change of the function \( f(x) = x^3 \) at \( x = -5 \) is 75.

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Most popular questions from this chapter

Let \(f(p)\) be the number of cars sold when the price is \(p\) dollars per car. Interpret the statements \(f(10,000)=200,000\) and \(f^{\prime}(10,000)=-3\).

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=-1+\frac{2}{x-2}\)

Each limit in Exercises 49-54 is a definition of \(f^{\prime}(a)\). Determine the function \(f(x)\) and the value of \(a\). \(\lim _{h \rightarrow 0} \frac{(1+h)^{-1 / 2}-1}{h}\)

Rates of Change Suppose that \(f(x)=4 x^{2}\). (a) What is the average rate of change of \(f(x)\) over each of the intervals 1 to 2,1 to \(1.5\), and 1 to \(1.1 ?\) (b) What is the (instantaneous) rate of change of \(f(x)\) when \(x=1 ?\)

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=x+\frac{1}{x}\)
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