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Chapter 4: Problem 36

Sketch the graph of the equation. $$y=e^{-1000 x}$$

Short Answer

Expert verified
The graph of \( y = e^{-1000x} \) is an exponentially decreasing curve, starting at (0, 1), hugging the x-axis for \( x > 0 \).

Step by step solution

01

Identify the Function Type

The given equation is \( y = e^{-1000x} \), which is an exponential decay function. The general form of an exponential function is \( y = ae^{bx} \). Here, \( a = 1 \) and \( b = -1000 \), indicating that it is an exponentially decreasing function.
02

Determine Key Characteristics

For \( y = e^{-1000x} \), as \( x \) increases, \( y \) exponentially approaches zero but never actually reaches it. At \( x = 0 \), \( y = e^{0} = 1 \). This gives us the y-intercept at point (0, 1). The graph decreases rapidly towards the x-axis because of the high negative exponent.
03

Sketch the Graph

Start by plotting the y-intercept at (0, 1). Since the function decreases quickly, draw a curve that moves downwards very sharply from the y-intercept and approaches the x-axis as \( x \) increases. The curve is close to the x-axis almost immediately after leaving the y-axis due to the large negative exponent.
04

Consider Asymptotic Behavior

The x-axis (or \( y = 0 \)) acts as a horizontal asymptote that the graph approaches but never touches. This means as \( x \rightarrow \infty \), \( y \rightarrow 0 \). Include this asymptotic behavior in your sketch by ensuring the graph gets very close to the x-axis.

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Key Concepts

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

Exponential Function
An exponential function is characterized by its formula: \( y = ae^{bx} \). Here, \( a \) is a constant that represents the initial amount when \( x = 0 \), and \( b \) affects the growth or decay rate. In our function \( y = e^{-1000x} \), \( a = 1 \) and \( b = -1000 \). Since \( b \) is negative, this represents exponential decay.

Exponential decay functions model situations where quantities decrease rapidly.
  • Think of radioactive decay or cooling of a liquid.
  • As \( x \) increases, \( e^{-1000x} \) rapidly approaches zero.
Rapid changes are typical in such functions due to the steepness introduced by the high negative exponent. Understanding the fundamental nature of an exponential function helps in predicting how it behaves over time.
Asymptotic Behavior
Asymptotic behavior describes how a function behaves as it moves towards infinity or a particular direction.

For \( y = e^{-1000x} \), as \( x \) becomes larger, the output value \( y \) gets very close to zero but never actually reaches it. This behavior forms a horizontal asymptote along the x-axis (or \( y = 0 \)).

Why does this happen? Because in the exponential function, the term \( e^{-1000x} \) decreases rapidly, approaching zero.

Important points to understand:
  • In practical terms, this means that no matter how large \( x \) gets, \( y \) will never be zero.
  • This tells us about the long-term behavior of the function, providing insight into how it behaves as part of a larger system.
Knowing about asymptotes helps in understanding limits and predicting trends in real-world situations.
Graph Sketching
Sketching a graph of an exponential decay function like \( y = e^{-1000x} \) involves recognizing certain fundamental aspects.

Start with the y-intercept, which is the point where the graph crosses the y-axis. For our function, when \( x = 0 \), \( y = 1 \). Thus, the y-intercept is at (0, 1).

To sketch:
  • Begin at the y-intercept (0, 1).
  • Draw the curve rapidly decreasing towards the x-axis.
  • The high negative exponent causes the graph to hug the x-axis quickly after the intercept.
Consider the horizontal asymptote: while the graph approaches the x-axis, it never touches or crosses it.

This visualization provides a clear and intuitive way to understand the rapid decay, allowing us to see how quickly \( y \) diminishes as \( x \) grows larger.

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Most popular questions from this chapter

(a) Graph \(f\) using a graphing utility. (b) Sketch the graph of \(g\) by taking the reciprocals of \(y\) -coordinates in (a), without using a graphing utility. $$f(x)=\frac{e^{x}+e^{-x}}{2} ; \quad g(x)=\frac{2}{e^{x}+e^{-x}}$$

Graph \(f\) and \(g\) on the same coordinate plane, and estimate the solution of the inequality \(f(x)>g(x)\). $$f(x)=3 \log _{4} x-\log x ; \quad g(x)=e^{x}-0.25 x^{4}$$

government receipts (in billions of dollars) for selected years are listed in the table. $$\begin{array}{|lcccc|}\hline \text { Year } & 1910 & 1930 & 1950 & 1970 \\\\\hline \text { Recelpts } & 0.7 & 4.1 & 39.4 & 192.8 \\\\\hline \text { Year } & 1980 & 1990 & 2000 \\\\\hline \text { Receipts }& 517.1 & 1032.0 & 2025.2 \\\\\hline\end{array}$$ (a) Let \(x=0\) correspond to the year \(1910 .\) Plot the data, together with the functions \(f\) and \(g\) : (1) \(f(x)=0.786(1.094)^{x}\) (2) \(g(x)=0.503 x^{2}-27.3 x+149.2\) (b) Determine whether the exponential or quadratic function better models the data. (c) Use your choice in part (b) to graphically estimate the year in which the federal government first collected S1 trillion.

Compound interest If \(\$ 1000\) is invested at a rate of \(7 \%\) per year compounded monthly, find the principal after (a) 1 month (b) 6 months (c) 1 year (d) 20 years

A country presently has coal reserves of 50 million tons. Last year 6.5 million tons of coal was consumed. Past years' data and population projections suggest that the rate of consumption \(R\) (in million tons/year) will increase according to the formula \(R=6.5 e^{0.02 t},\) and the total amount \(T\) (in million tons) of coal that will be used in \(t\) years is given by the formula \(T=325\left(e^{0.02 t}-1\right) .\) If the country uses only its own resources, when will the coal reserves be depleted?
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