Problem 56 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. If \(y=A b^{x}\) and increasing \(x\) by 1 increases \(y\) by a factor of \(3,\) then increasing \(x\) by 2 increases \(y\) by a factor of 9

Step-by-Step Solution

Answer

True, increasing \( x \) by 2 multiplies \( y \) by 9.

Step 1: Understand the exponential function relationship

The function given is an exponential function: \( y = A b^x \). This means that \( y \) is determined by the base \( b \) raised to the power of \( x \), multiplied by the constant \( A \). Our task is to determine the change in \( y \) when \( x \) is increased.

Step 2: Analyze the initial condition

The problem states that increasing \( x \) by 1 multiplies \( y \) by a factor of 3. This means that \( y(x+1) = 3 \times y(x) \). In mathematical terms, this translates to: \[ Ab^{x+1} = 3 imes Ab^x \]. Simplifying, we find that \( b = 3 \).

Step 3: Determine the effect of increasing \( x \) by 2

Now, we need to calculate the factor by which \( y \) changes when \( x \) is increased by 2. Starting with \( y(x+2) = Ab^{x+2} \), we can express it as \( Ab^x \times b^2 \), or \( Ab^x imes 3^2 \), since \( b = 3 \). Therefore, \( y(x+2) = 9 imes y(x) \).

Step 4: Conclusion

The calculation confirms that increasing \( x \) by 2 multiplies \( y \) by a factor of 9. Therefore, if increasing \( x \) by 1 multiplies \( y \) by 3, then increasing \( x \) by 2 indeed multiplies \( y \) by 9.

Key Concepts

Exponential Growth

Exponential growth is all about how things increase exponentially, which means they become larger at a constantly increasing rate. This differs from linear growth, where increases happen at a constant rate. To understand exponential growth better, imagine the function given in the exercise, where \( y = A b^x \).

In such a function, as your \( x \) grows, the effect on \( y \) isn't just an add-on, it's a multiplier! The specific growth pattern described involves multiplying the original value rather than adding a fixed amount. This makes exponential growth appear rapid. A classic example is compound interest, where your money multiplies over time rather than just adding a fixed interest.

• Exponential growth can make numbers skyrocket quickly.
• Each increase in \( x \) delivers a bigger impact on \( y \).
• This is why we find it in finance, biology, and many technological contexts.

Understanding how these exponential functions work gives new insights into rapid changes and expansions in various fields.

Mathematical Reasoning

Mathematical reasoning is critical in making sense of mathematical statements and steps. In the given exercise, we need to logically connect what happens to \( y \) when \( x \) changes. We do this by carefully following the relationships and transformations intrinsic to the function \( y = A b^x \).

The task is to determine how each unit change in \( x \) affects \( y \). With mathematical reasoning, you know that if increasing \( x \) by 1 makes \( y \) three times bigger, increasing \( x \) by 2 should increase \( y \) by \( 3^2 = 9 \).

Here's a quick view of how logical steps look:

• Recognizing the given condition and translating it mathematically.
• Checking base impact: increasing \( x \) by 1 gives a factor of 3, meaning \( b = 3 \).
• Using this to reason out the effect of further increasing \( x \) by 2.
• Concluding based on these logical transitions.

Using clear reasoning helps simplify complex ideas, ensuring the results are both accurate and understandable.

Function Analysis

Understanding functions involves analyzing how changes in one variable affect another. In exponential functions like \( y = A b^x \), your job is to see the pattern and quantify changes as inputs change. Specifically, the challenge is to predict outputs effectively by analyzing how \( y \) behaves as a function of \( x \).

The problem here is an example of how small changes in \( x \) can significantly impact \( y \) because of the exponential nature of the function. Some steps to analyze functions effectively include:

• Grasping the general formula and its components—here, \( A \) is the coefficient, \( b \) represents the base.
• Looking at the initial condition—a unit change causing a threefold increase informs us that \( b = 3 \).
• Determining impacts through mathematical calculations, building confidence in predicting outcomes for larger changes.
• Observing these patterns improves understanding of growth trends applicable in various realms.

Proper function analysis lets you master the relationships between variables, turning complex patterns into manageable pieces of information.