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Chapter 5: Problem 27

Find the average rate of change of the function. \(g(x)=3^{x^{2}-x-3}\) as \(x\) goes from -1 to 1

Short Answer

Expert verified
The average rate of change of the function g(x) as x goes from -1 to 1 is \(-\frac{4}{27}\).

Step by step solution

01

Evaluate the function at x = -1 and x = 1

We need to find the value of the g(-1) and g(1). \[g(-1) = 3^{(-1)^{2} - (-1) - 3}\] \[g(-1) = 3^{-1}\] \[g(1) = 3^{(1)^{2} - 1 - 3}\] \[g(1) = 3^{-3}\]
02

Apply the formula for average rate of change

The average rate of change of the function g(x) is given by the formula: \[\frac{g(b) - g(a)}{b - a}\] Here, a = -1 and b = 1. Plug in the function values we calculated in Step 1: \[\frac{g(1) - g(-1)}{1 - (-1)} = \frac{3^{-3} - 3^{-1}}{1 - (-1)}\]
03

Simplify the expression

Now, simplify the expression to find the average rate of change: \[\frac{3^{-3} - 3^{-1}}{2} = \frac{1/27 - 1/3}{2}\] \[\frac{(1 - 9)/27}{2} = \frac{-8/27}{2}\] \[-\frac{8}{27} \cdot \frac{1}{2} = -\frac{8}{54} = -\frac{4}{27}\] So, the average rate of change of the function \(g(x) = 3^{x^{2} - x - 3}\) as \(x\) goes from -1 to 1 is \(-\frac{4}{27}\).

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Key Concepts

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

Exponential Functions
Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. In the function given,
  • The base is 3, and the exponent is a quadratic expression: \( x^2 - x - 3 \).
  • This means the value of the function changes rapidly depending on the value of \( x \).
These functions are distinct from linear or polynomial functions due to their rate of growth, which can be much faster. Exponential functions are often used to model real-world phenomena such as population growth or radioactive decay, where changes happen swiftly and by multiplying factors. In essence, the power of the exponent dictates the behavior and rate of increase or decrease within the function.
Function Evaluation
Function evaluation involves finding the value of a function at a specific point. For the function
  • \( g(x) = 3^{x^2 - x - 3} \),
    we need to evaluate it at particular values of \( x \) to find \( g(-1) \) and \( g(1) \).
  • By substituting \( -1 \) for \( x \), we get \( g(-1) = 3^{(-1)^2 - (-1) - 3} = 3^{-1} \).
  • Similarly, for \( x = 1 \), \( g(1) = 3^{1^2 - 1 - 3} = 3^{-3} \).
Function evaluation is crucial for determining specific values of a function, which in turn helps us understand how the function behaves in different contexts. Checking these values tells us how the function outputs vary as inputs change.
Simplification
Simplification in mathematics involves reducing expressions to a simpler form, making calculations easier. Let's simplify the expression for the average rate of change.Given the expression:
  • \( \frac{3^{-3} - 3^{-1}}{2} \), we first rewrite the terms:
    \( 3^{-3} \) means \( \frac{1}{27} \) and \( 3^{-1} \) means \( \frac{1}{3} \).
Performing the subtraction:
  • \( \frac{1}{27} - \frac{1}{3} = \frac{1 - 9}{27} = \frac{-8}{27} \).
Finally, divide the result by 2:
  • \( \frac{-8/27}{2} = -\frac{8}{54} \), which simplifies further to \( -\frac{4}{27} \).
Simplification helps us present a clear and more manageable version of complex calculations, making it easier to find final results.
Rate of Change Formula
The rate of change formula is commonly used to determine how one quantity changes in relation to another. The formula can be written as:
  • \( \frac{g(b) - g(a)}{b - a} \), where \( g(b) \) and \( g(a) \) are function values at points \( b \) and \( a \) respectively.
For our function, the average rate of change calculation is:
  • \( \frac{g(1) - g(-1)}{1 - (-1)} = \frac{3^{-3} - 3^{-1}}{2} = -\frac{4}{27} \).
This formula helps us understand the average change of a function over a specific interval.
It's a crucial concept for analyzing the behavior of functions, ensuring we have an idea of their overall tendencies over a set of values.

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Most popular questions from this chapter

According to one theory of learning, the number of words per minute \(N\) that a person can type after \(t\) weeks of practice is given by \(N=c\left(1-e^{-k t}\right),\) where \(c\) is an upper limit that \(N\) cannot exceed and \(k\) is a constant that must be determined experimentally for each person. (a) If a person can type 50 wpm (words per minute) after four weeks of practice and 70 wpm after eight weeks, find the values of \(k\) and \(c\) for this person. According to the theory, this person will never type faster than \(c\) wpm. (b) Another person can type 50 wpm after four weeks of practice and 90 wpm after eight weeks. How many weeks must this person practice to be able to type 125 wpm?

The spread of a flu virus in a community of 45,000 people is given by the function $$f(t)=\frac{45,000}{1+224 e^{-.899 t}}$$ where \(f(t)\) is the number of people infected in week \(t\). (a) How many people had the flu at the outbreak of the epidemic? After three weeks? (b) When will half the town be infected?

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). (a) How long will it take to triple your money if you invest 500 dollars at a rate of \(5 \%\) per year compounded annually? (b) How long will it take at \(5 \%\) compounded quarterly?

Rationalize the denominator and simplify your answer. $$\frac{1+\sqrt{3}}{5+\sqrt{10}}$$

Sketch a complete graph of the function. $$h(x)=(1 / e)^{-x}$$
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