/*! 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 114 Sketch the graph of a function w... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 114

Sketch the graph of a function whose derivative is exactly 1 at every point

Short Answer

Expert verified
The function with a derivative of 1 at every point is a linear function of the form \(f(x) = x + b\), where b is any constant. For simplicity, we can set the y-intercept to 0, resulting in the function \(f(x) = x\). The graph of this function is a straight line passing through the origin (0,0) and making a 45-degree angle with the x-axis.

Step by step solution

01

Find the function with derivative 1 at every point

Let f(x) be the desired function we want to find. We know that the derivative of f(x), f'(x), is 1 at every point. Since the derivative represents the slope of the tangent line at a point, this means that the slope of f(x) is constant and equal to 1. When the slope of a function is constant, it suggests that the function is linear. A linear function is of the form: \[f(x) = mx + b\] where m is the slope and b is the y-intercept. Since the slope, m, is equal to 1, the function f(x) is simply: \[f(x) = x + b\]
02

Determine the y-intercept

Since we are not given any specific points on the graph of the function, we can set the y-intercept 'b' to any constant value. For simplicity, we can set the y-intercept to 0. This means our function will have the form: \[f(x) = x\]
03

Sketch the graph of the function

We now have the function \(f(x) = x\), which is a simple linear function with slope equal to 1 and y-intercept 0. To sketch the graph, begin by plotting the y-intercept (0,0) on the Cartesian plane. Now, since the slope is 1, for every 1 unit increase in x, there is also a 1 unit increase in y. You can draw the function by plotting points and then connecting them to form a straight line. The graph of the function f(x) = x is a straight line passing through the origin (0,0) and making a 45-degree angle with the x-axis. This line extends infinitely in both positive and negative x directions. Now you have successfully sketched the graph of a function whose derivative is exactly 1 at every point.

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Key Concepts

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

Linear function
A linear function is one of the simplest types of functions in mathematics. It involves only first-degree equations, meaning the highest power of the variable is one. Such functions are described by the formula:
  • \(f(x) = mx + b\)
where:
  • \(m\) is the slope of the line, showing how steep it is and in which direction it rises or falls.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis on a graph.
In the context of the exercise, we identified that the derivative of the function is a constant, which implies it's a linear function. Since the derivative is 1, the function is linearly increasing, mirroring the form \(f(x) = x + b\), where any changes in \(b\) will shift the line up or down but will not affect the slope. Understanding this fundamental concept simplifies problem-solving in calculus.
Slope
The slope is a crucial concept in understanding functions and their derivatives. Geometrically, it can be visualized as the measure of steepness or the degree of tilt of a straight line. For a linear function described by:
  • \(f(x) = mx + b\)
The slope \(m\) is the coefficient of \(x\). It dictates how much \(y\) changes as \(x\) changes. A slope of 1, as seen in this exercise, indicates that for every one unit you move horizontally to the right (increase in \(x\)), you move one unit vertically up (increase in \(y\)).
This creates a perfect diagonal line that makes a 45-degree angle with the x-axis. In our case, \(f'(x) = 1\) everywhere means the line is uniformly increasing at a constant rate. This consistency simplifies both understanding and drawing the graph of the function.
Graph sketching
Graph sketching is a foundational skill in visualizing mathematical functions. It helps in understanding the behavior of functions at a glance. For a linear function like \(f(x) = x\), sketching starts with the identification of key features:
  • The y-intercept, the point where the graph crosses the y-axis. In \(f(x) = x\), this is at (0,0).
  • The slope, which tells us the direction and steepness of the line. Here, a slope of 1 means for each step right on the x-axis, you step up equally on the y-axis.
To graph it, start at the y-intercept, then use the slope to plot the next points. Connect these points with a straight line extending infinitely in both directions. The line should make a 45-degree angle with the x-axis, reflecting the uniform rate of increase defined by the slope. Mastering graph sketching, especially for straightforward formulas, can vastly improve comprehension of more complex functions.

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Most popular questions from this chapter

In the early stages of the deadly SARS (Severe Acute Respiratory Syndrome) epidemic in 2003, the number of reported cases can be approximated by $$ A(t)=167(1.18)^{t} \quad(0 \leq t \leq 20) $$ \(t\) days after March 17, 2003 (the first day in which statistics were reported by the World Health Organization). a. What, approximately, was the instantaneous rate of change of \(A(t)\) on March \(27(t=10) ?\) Interpret the result. b. Which of the following is true? For the first 20 days of the epidemic, the instantaneous rate of change of the number of cases (A) increased (B) decreased (C) increased and then decreased (D) decreased and then increased

Draw the graph of a function \(f\) with the property that the balanced difference quotient gives a more accurate approximation of \(f^{\prime}(1)\) than the ordinary difference quotient.

Daily oil production by Pemex, Mexico's national oil company, can be approximated by \(P(t)=-0.022 t^{2}+0.2 t+2.9\) million barrels \(\quad(1 \leq t \leq 9)\) where \(t\) is time in years since the start of \(2000 .^{59}\) Find the derivative function \(\frac{d P}{d t} .\) At what rate was oil production changing at the start of \(2004(t=4) ?\) HINT [See Example 4.]

In Exercises 39-44, find the equation of the tangent to the graph at the indicated point. HINT [Compute the derivative algebraically; then see Example \(2(\mathrm{~b})\) in Section \(10.5 .]\) $$ f(x)=x^{2}-3 ; a=2 $$

Estimate the given quantity. \(f(x)=2 e^{x} ;\) estimate \(f^{\prime}(1)\)
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