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Chapter 3: Problem 27

Find the instantaneous rates of change of the given functions at the indicated points. \(f(x)=x^{2}+2 x+3, c=-1\)

Short Answer

Expert verified
The instantaneous rate of change at \( x = -1 \) is 0.

Step by step solution

01

Understand the Problem

We need to find the instantaneous rate of change of the function \( f(x) = x^2 + 2x + 3 \) at \( x = c = -1 \), which is the same as finding the derivative of the function at \( x = -1 \).
02

Find the Derivative of the Function

The first step in finding the instantaneous rate of change is to find the derivative of the function. The derivative, \( f'(x) \), of the function \( f(x) = x^2 + 2x + 3 \), can be found using standard differentiation rules. 1. The derivative of \( x^2 \) is \( 2x \).2. The derivative of \( 2x \) is \( 2 \).3. The derivative of a constant \( 3 \) is \( 0 \). Therefore, \[ f'(x) = 2x + 2. \]
03

Evaluate the Derivative at the Indicated Point

Now we substitute \( x = -1 \) into the derivative to find the instantaneous rate of change at that point. \[ f'(-1) = 2(-1) + 2 = -2 + 2 = 0. \]
04

Interpret the Result

The value \( f'(-1) = 0 \) means that the instantaneous rate of change of the function \( f(x) = x^2+2x+3 \) at \( x = -1 \) is 0. This suggests that at \( x = -1 \), the slope of the tangent to the curve is horizontal.

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Key Concepts

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

Differentiation
Differentiation is a fundamental concept in calculus. It helps us understand how a function changes at any given point. Imagine differentiation as a tool that lets us find the rate at which a function is increasing or decreasing. This is especially useful in fields like physics, economics, and biology. Here, we are looking for the instantaneous rate of change of a function.To differentiate a function, we apply specific rules. For example:
  • The power rule: If you have a function like \(x^n\), its derivative is \(nx^{n-1}\).
  • The constant rule: The derivative of a constant is always 0.
  • The sum rule: The derivative of a sum of functions is the sum of their derivatives.
In our exercise, we apply these rules to find the derivative of \(f(x) = x^2 + 2x + 3\). We end up with \(f'(x) = 2x + 2\), which tells us the rate of change of the function with respect to \(x\).
Derivative Evaluation
Once we have found the derivative of a function, the next step is to evaluate it at a specific point. This gives us the instantaneous rate of change of the function at that point. In simpler terms, it tells us how steep or flat the function graph is at a given value of \(x\).In the exercise, the derivative \(f'(x) = 2x + 2\) is evaluated at \(x = -1\). We plug \(-1\) into the equation:\[f'(-1) = 2(-1) + 2 = -2 + 2 = 0\]This calculation shows that the rate of change is 0. So, at \(x = -1\), the graph of the original function is not sloping up or down; it's flat.
Slope of Tangent
The slope of the tangent line at a particular point on a curve is a physical representation of the derivative at that point. It tells us how the curve behaves at that exact spot.When you evaluate the derivative, as in our case where \(f'(-1) = 0\), you are finding the slope of the tangent line to the curve of the function at \(x = -1\). A zero slope means the tangent line is horizontal, which we interpret as a point where the function is neither increasing nor decreasing.Understanding the slope of the tangent is vital in many practical applications. For example:
  • In physics, it might explain the velocity of an object at a particular time.
  • In economics, it could foretell profit or loss trends at a specific production level.
  • In science, it helps determine maximum and minimum points, crucial for optimization.
Thus, finding the slope of the tangent is more than just a calculation; it's a tool to understand the behavior of functions.

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Most popular questions from this chapter

Environmental Entomology Allen and coworkers \(^{40}\) found the relationship between the temperature and the number of eggs laid by the female citrus rust mite. They found that if \(x\) is the temperature in degrees Celsius and \(y\) is the total eggs per female, \(y\) was given approximately by the equation \(-0.00574 x^{3}+0.292 x^{2}-3.632 x+11.661\) Graph using a window with dimensions [10,38.2] by \([0,20] .\) Have your computer or graphing calculator draw tangent lines to the curve when \(x\) is \(19,23.5,25,28,\) and 31\. Note the slope and relate this to the rate of change. Interpret what each of these numbers means. On the basis of this model, describe what happens to the rate of change of number of eggs as temperature increases.

In Exercise 67 of Section 3.3 you found that \(d / d x(\sin x)=\) \(\cos x .\) Notice from your computer or calculator that \(\sin 0=\) 0 and \(\cos 0=1\). Use all of this and the tangent line approximation to approximate \(\sin 0.02\). Compare your answer to the value your computer or calculator gives.

Biology An anatomist wishes to measure the surface area of a bone that is assumed to be a cylinder. The length \(l\) can be measured with great accuracy and can be assumed to be known exactly. However, the radius of the bone varies slightly throughout its length, making it difficult to decide what the radius should be. Discuss how slight changes in the radius \(r\) will approximately affect the surface area.

Find, without graphing, where each of the given functions is continuous. $$ f(x)=\left\\{\begin{array}{ll} -x+1 & \text { if } x<0 \\ x^{2} & \text { if } x \geq 0 \end{array}\right. $$

Economic Growth and the Environment Grossman and Krueger \(^{41}\) studied the relationship in a variety of countries between per capita income and various environmental indicators. The object was to determine whether environmental quality deteriorates steadily with growth. They found that the equation $$ y=f(x)=0.27 x^{3}-6.31 x^{2}+39.87 x+34.78 $$ approximated the relationship between \(x\) given as GDP per capita income in thousands of dollars and \(y\) given as units of smoke in cities. Graph on your computer or graphing calculator using a screen with dimensions [0,9.4] by \([0,150] .\) Have your computer or graphing calculator draw tangent lines to the curve when \(x\) is \(2,3,5,\) and \(6 .\) Note the slope, and relate this to the rate of change. Interpret what each of these numbers means. On the basis of this model. determine whether environmental quality deteriorates with economic growth.
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