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Chapter 4: Problem 28

Suppose that \(f(0)=5\) and that \(f^{\prime}(x)=2\) for all \(x .\) Must \(f(x)=\) \(2 x+5\) for all \(x ?\) Give reasons for your answer.

Short Answer

Expert verified
Yes, \(f(x)=2x+5\) for all \(x\).

Step by step solution

01

Consider the Information Given

We are given that \( f(0) = 5 \) and \( f'(x) = 2 \) for all \( x \). This tells us that the function \( f(x) \) has a constant derivative of 2, which means it is a linear function where the slope is 2.
02

Set Up the General Equation

Since the derivative is constant \( f'(x) = 2 \), the function \( f(x) \) must be of the form \( f(x) = 2x + C \), where \( C \) is a constant that we need to determine.
03

Solve for the Constant \(C\)

Use the information \( f(0) = 5 \) to find \( C \). Substituting \(x = 0\) into \( f(x) = 2x + C \), we have \( f(0) = 2(0) + C = 5 \). This simplifies to \( C = 5 \).
04

Write the Specific Equation

Having found \( C = 5 \), substitute back into the equation \( f(x) = 2x + C \) to get the specific function \( f(x) = 2x + 5 \).
05

Confirm Consistency

Check if \( f'(x) \) for \( f(x) = 2x + 5 \) equals 2 for all \( x \). Taking the derivative, \( f'(x) = 2 \), which matches the given \( f'(x) = 2 \). Therefore, \( f(x) = 2x + 5 \) is consistent with all given information.

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Key Concepts

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

Derivative
In calculus, the derivative of a function represents how the function's output changes as its input changes. It's essentially the "rate of change" or "slope" of the function at any given point. For a linear function, the derivative is a constant, matching the slope of the line. For example, if you have a function like \( f(x) = 2x + 5 \), its derivative is simply \( f'(x) = 2 \).
This tells us that for every 1 unit increase in \(x\), \(f(x)\) increases by 2 units.
In this scenario, knowing that \( f'(x) = 2 \) for all \( x \) confirms that \( f(x) \) must be linear. The consistent derivative across all values of \( x \) indicates that the slope of the function never changes.
  • The derivative tells us how steep the function is.
  • For linear functions, the derivative is always the same.
  • A constant derivative implies a straight line.
Linear Function
A linear function is one of the simplest types of functions you'll encounter in mathematics. Its general form is \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In the case of our problem, \( f(x) = 2x + 5 \) is a perfect example of a linear function with \( m = 2 \) and \( b = 5 \).
Some characteristics of linear functions include:
  • They form straight lines when graphed.
  • The slope \( m \) dictates how steep the line is.
  • The y-intercept \( b \) shows where the line crosses the y-axis.
Understanding that the constant derivative \( f'(x) = 2 \) implies a linear function allows us to quickly identify its form. Here, since the line has a slope of 2, the original function must include that slope along with the constant \( C \), which we determined using given information.
Constant
Constants in functions are values that do not change; they remain the same no matter what value the input \(x\) takes. In our exercise, the constant \( C \) was determined using the condition \( f(0) = 5 \).
Let's explore constants further:
  • In a linear function like \( f(x) = 2x + C \), the \( C \) value indicates the y-intercept.
  • For \( f(0) = 5 \), substituting \( x = 0 \) into \( f(x) = 2x + C \) results in \( C = 5 \).
  • This means the function intersects the y-axis at 5.
In summary, constants are crucial in shaping the position of a function on a graph. In this instance, \( C = 5 \) not only satisfies the original condition but also completes our linear function, resulting in \( f(x) = 2x + 5 \).
It ensures that the function correctly represents the given information and maintains its structure across all \( x \) values.

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Most popular questions from this chapter

Suppose that the second derivative of the function \(y=f(x)\) is $$y^{\prime \prime}=x^{2}(x-2)^{3}(x+3).$$ For what \(x\)-values does the graph of \(f\) have an inflection point?

Wilson lot size formula One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is $$A(q)=\frac{k m}{q}+c m+\frac{h q}{2},$$ where \(q\) is the quantity you order when things run low (shoes, radios, brooms, or whatever the item might be \(, k\) is the cost of placing an order (the same, no matter how often you order), \(c\) is the cost of one item (a constant), \(m\) is the number of items sold each week (a constant), and \(h\) is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security). \begin{equation} \begin{array}{l}{\text { a. Your job, as the inventory manager for your store, is to find }} \\ \quad {\text { the quantity that will minimize } A(q) . \text { What is it? (The formula }} \\ \quad {\text { you get for the answer is called the Wilson lot size formula.) }} \\ {\text { b. Shipping costs sometimes depend on order size. When they }} \\ \quad {\text { do, it is more realistic to replace } k \text { by } k+b q \text { , the sum of } k} \\\ \quad {\text { and a constant multiple of } q . \text { What is the most economical }} \\ \quad {\text { quantity to order now? }}\end{array}\end{equation}

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x^{2 / 5}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\left|x^{2}-1\right|$$

\begin{equation} \begin{array}{l}{\text { a. The function } y=\cot x-\sqrt{2} \csc x \text { has an absolute maxi mum }} \\ \quad {\text { value on the interval } 0< x <\pi . \text { Find it. }} \\ {\text { b. Graph the function and compare what you see with your }} \\ \quad {\text { answer in part (a). }}\end{array} \end{equation}
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