/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 Find \(f^{\prime}(x)\). $$f(x)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 46

Find \(f^{\prime}(x)\). $$f(x)=-0.01 x^{2}+0.4 x+50$$

Short Answer

Expert verified
-0.02x + 0.4

Step by step solution

01

- Identify the function

The given function is: \[ f(x) = -0.01 x^{2} + 0.4 x + 50 \] This is a quadratic function.
02

- Apply the power rule for differentiation

To find the derivative, use the power rule which states that for any term of the form \( ax^n \), the derivative is \( n \times ax^{n-1} \).
03

- Differentiate each term

Differentiate each term of the function individually: \[ \text{First term: } -0.01 x^{2} \rightarrow -0.01 \times 2 \times x^{2-1} = -0.02 x \] \[ \text{Second term: } 0.4 x \rightarrow 0.4 \times 1 \times x^{1-1} = 0.4 \] The constant term 50 has a derivative of 0.
04

- Combine the derivatives

Combine the differentiated terms to get the final result: \[ -0.02 x + 0.4 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Quadratic Functions
Quadratic functions are a type of polynomial function where the highest degree term is squared. For example, in the function \(f(x)=-0.01x^2 + 0.4x + 50\), the term \(-0.01x^2\) is the quadratic term.
This function can be written in the general form \(ax^2 + bx + c\), where:\
  • \(a\), \(b\), and \(c\) are constants
  • \(a\) is not equal to zero
Quadratic functions create U-shaped graphs called parabolas when plotted on a coordinate plane.
If \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, it opens downwards.
In our case, \(a\) is \(-0.01\), which means the parabola will open downwards.
Power Rule
The power rule is one of the foundational rules in calculus for finding derivatives. It states that if you have a term \(ax^n\), its derivative is \(n \times ax^{n-1}\).
Let's break this down with the given function \(f(x)=-0.01x^2 + 0.4x + 50\).
  • For the term \(-0.01x^2\), apply the power rule: multiply the exponent by the coefficient and reduce the exponent by 1:
  • For the term \(0.4x\), the exponent of \(x\) is 1. Using the power rule, the result is simply the coefficient \(0.4\).
  • The constant term \(50\) has a power of 0, and its derivative is always 0.
So, using the power rule, you're able to find the derivative of polynomial terms easily and systematically.
Derivative Calculations
Derivative calculations help us understand the rate at which a function changes. For example, in the function \(f(x)=-0.01x^2 + 0.4x + 50\), calculating the derivative provides a new function that tells us the slope of the original function at any point.
  • First, identify each term separately: \(-0.01x^2\), \(0.4x\), and \(50\).
  • Apply the power rule to each term:


\(-0.01x^2\): The power rule gives us \(-0.01 \times 2 \times x^{2-1} = -0.02x\)
\(0.4x\): The power rule gives us \(0.4 \times 1 \times x^{1-1} = 0.4\)
The constant term \(50\) drops out because its derivative is 0.

Now, combine these individual derivatives to get the final result:
\(f^{\rightarrow}(x) = -0.02x + 0.4\)
This new function \(-0.02x + 0.4\) represents the rate of change of the original function \(f(x)\).

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use a graphing calculator to check the results of Exercises \(1-48\).

Differentiate. $$y=\left(\frac{x^{2}-x-1}{x^{2}+1}\right)^{3}$$

Find \(d y / d x .\) Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$y=(x+3)(x-2)$$

The population \(P\), in thousands, of a small city is given by \(P(t)=\frac{500 t}{2 t^{2}+9}\) where \(t\) is the time, in years. (GRAPH CANNOT COPY) a) Find the growth rate. b) Find the population after 12 yr. c) Find the growth rate at \(t=12\) yr.

Business profit. A company is selling laptop computers. It determines that its total profit, in dollars, is given by \(P(x)=0.08 x^{2}+80 x\) where \(x\) is the number of units produced and sold. Suppose that \(x\) is a function of time, in months, where \(x=5 t+1\) a) Find the total profit as a function of time \(t\) b) Find the rate of change of total profit when \(t=48\) months.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.