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Chapter 2: Problem 56

If you zoom in (with your calculator) on the graph of \(y=f(x)\) in a small interval around \(x=10\) and see a straight line, then the slope of that line equals the derivative \(f^{\prime}(10)\).

Short Answer

Expert verified
The slope of the line is the derivative at that point: \( f'(10) \).

Step by step solution

01

Understand the Problem

The problem states that when we zoom in closely on the graph of \( y = f(x) \) around \( x = 10 \) and observe a straight line, the slope of that line is the derivative \( f'(10) \). We need to determine what this graphical behavior implies.
02

Identify the Graphical Representation

Zooming in on a graph until it appears as a straight line means we are observing the tangent to the function at that point. This is a way of visualizing the derivative, as the tangent line represents the instantaneous rate of change of the function at a particular point.
03

Connect Graphical Representation to Derivative

The slope of the tangent line at a point on the graph of a function is defined as the derivative of the function at that point. Therefore, when the graph appears as a straight line near \( x = 10 \), its slope is \( f'(10) \).
04

Conclude with the Analytical Definition

The derivative \( f'(x) \) at a specific point \( x = a \) is calculated by the limit: \( f'(a) = \lim_{{h \to 0}} \frac{f(a+h) - f(a)}{h} \). If the graph straightens to a line as described, this corresponds to \( f'(10) \), the slope of that line.

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Key Concepts

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

Tangent Line
A tangent line is a straight line that just "touches" a curve at a particular point. This line provides an approximation of the curve's behavior right at that specific point. When you zoom in really close on a curve around a point, the curve may look just like a line. That line is the tangent line.
  • It gives the best linear approximation to the function near the specific point.
  • Only touches the curve at that one point of tangency.
  • Shows the direction in which the curve moves.
Visualizing the tangent line helps us understand how steep the curve is at that exact place on the graph. At the point of tangency, the tangent line mirrors the curve's behavior perfectly.
Slope
The term "slope" refers to how steep a line is. Mathematically, slope is expressed as "rise over run," or the ratio of vertical movement to horizontal movement between two points. In the context of a tangent line to a curve, the slope is crucial because:
  • It tells us how fast the function's value (y-coordinate) is changing at a specific point (x-coordinate).
  • A greater slope means a steeper incline or decline.
  • A positive slope indicates an upward trend, while a negative slope indicates a downward trend.
For the tangent line mentioned in the exercise, its slope at the point is exactly the same as the derivative of the function at that point. So if you see a straight line while zooming in closely on a curve, its slope equals the derivative.
Instantaneous Rate of Change
Instantaneous rate of change is a critical concept that tells us how a quantity is changing at a specific moment. This is somewhat like looking at the speedometer of a car to see how fast you're going at a single point in time. Rather than calculating across an interval, it measures the rate at a single point.
  • Equivalent to the slope of the tangent line for a function at a given point.
  • Gives the exact rate at which the function value is altering.
  • In calculus, it is represented by the derivative.
When the exercise mentions observing a straight line by zooming in near a point, it describes how that slope embodies the instantaneous rate of change of the function at that point.
Limit Definition of Derivative
The limit definition of a derivative provides a formal method for finding the derivative, or instantaneous rate of change, of a function at any given point. It is denoted as:\[ f'(a) = \lim_{{h \to 0}} \frac{f(a+h) - f(a)}{h}\]This definition helps us calculate the derivative precisely:
  • "f(a+h) - f(a)" represents the change in function values as you move from "a" to "a+h".
  • "h" is the small change in the input value, approaching zero.
  • The limit process is used to find the exact rate of change as the interval becomes infinitesimally small.
In the exercise context, when the graph looks like a straight line near \( x = 10 \), the derivative \( f'(10) \) is determined by this limit method, explaining the slope of the tangent line at that point.

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Most popular questions from this chapter

Suppose \(P(t)\) is the monthly payment, in dollars, on a mortgage which will take \(t\) years to pay off. What are the units of \(P^{\prime}(t) ?\) What is the practical meaning of \(P^{\prime}(t) ?\) What is its sign?

Draw the graph of a continuous function \(y=f(x)\) that satisfies the following three conditions: - \(f^{\prime}(x)>0\) for \(13\) \(f^{\prime}(x)=0\) at \(x=1\) and \(x=3\)

An economist is interested in how the price of a certain commodity affects its sales. Suppose that at a price of $$ p\( a quantity \)q\( of the commodity is sold. If \)q=f(p),\( explain in economic terms the meaning of the statements \)f(10)=240,000\( and \)f^{\prime}(10)=-29,000.$

On May \(9,2007,\) CBS Evening News had a 4.3 point rating. (Ratings measure the number of viewers.) News executives estimated that a 0.1 drop in the ratings for the CBS Evening News corresponds to a 5.5 million dollars drop in revenue. \(^{6}\) Express this information as a derivative. Specify the function, the variables, the units, and the point at which the derivative is evaluated.

During the 1970 s and 1980 s, the build up of chlorofluorocarbons (CFCs) created a hole in the ozone layer over Antarctica. After the 1987 Montreal Protocol, an agreement to phase out CFC production, the ozone hole has shrunk. The ODGI (ozone depleting gas index) shows the level of CFCs present. \(^{8}\) Let \(O(t)\) be the ODGI for Antarctica in year \(t ;\) then \(O(2000)=95\) and \(O^{\prime}(2000)=\) -1.25. Assuming that the ODGI decreases at a constant rate, estimate when the ozone hole will have recovered, which occurs when ODGI = 0.
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