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Chapter 5: Problem 5

Find \(C\) and \(a\) such that the function \(f(x)=C a^{x}\) satisfies the given conditions. a. \(f(0)=6\) and for each unit increase in \(x,\) the output is multiplied by 1.2 . b. \(f(0)=10\) and for each unit increase in \(x,\) the output is multiplied by 2.5

Short Answer

Expert verified
For (a): \(C = 6\) and \(a = 1.2\). For (b): \(C = 10\) and \(a = 2.5\).

Step by step solution

01

Identify Initial Values

For the function of the form \(f(x) = C a^x\), use the initial condition \(f(0) = C \).
02

Set the Initial Condition for Part (a)

Given \(f(0) = 6\), therefore \(C = 6\).
03

Derive the Base for Part (a)

Given that for each unit increase in \(x\), the output multiplies by 1.2, we can say that \(a = 1.2\).
04

Write the Function for Part (a)

Thus, the function is \(f(x) = 6 \cdot 1.2^x\).
05

Set the Initial Condition for Part (b)

Given \(f(0) = 10\), therefore \(C = 10\).
06

Derive the Base for Part (b)

Given that for each unit increase in \(x\), the output multiplies by 2.5, we can say that \(a = 2.5\).
07

Write the Function for Part (b)

Thus, the function is \(f(x) = 10 \cdot 2.5^x\).

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Key Concepts

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

Base of an Exponential Function
In the context of exponential functions, the base is a crucial component. It determines how the function behaves as the independent variable changes. In the general form of an exponential function, which is expressed as: \[f(x) = C \times a^x\] the variable \(a\) represents the base. The base dictates the growth rate or decay rate depending on its value. If \(a > 1\), the function represents exponential growth. Conversely, if \(0 < a < 1\), it indicates exponential decay.
For instance, in the problem provided:
  • In part (a), the function has a base of 1.2, representing an exponential growth because 1.2 is greater than 1.
  • In part (b), the function has a base of 2.5, also indicating exponential growth.
These bases show how rapidly the function values change as \(x\) increases. A higher base means the values shoot up faster.
Initial Value
The initial value of an exponential function is the value of the function when the independent variable \(x\) is zero. It is represented by the coefficient \(C\) in the exponential function's standard form: \[f(x) = C \times a^x\] The initial value can be interpreted as the starting amount before any growth or decay begins.
In the exercise, this is reflected as:
  • For part (a), the initial value is 6. This means when \(x = 0\), the function \(f(x) = 6 \times 1.2^0\) is 6.
  • For part (b), the initial value is 10. So, when \(x = 0\), the function \(f(x) = 10 \times 2.5^0\) is 10.
The initial value essentially provides the starting point for the function from which the exponential changes take place.
Function Transformations
Exponential functions can be transformed in various ways to suit different scenarios. Transformations affect the function's graph, influencing its position, shape, and orientation. Here are key transformation types in the context of exponential functions:
  • Vertical Shifts: Adjusting the function by adding or subtracting a constant (e.g., \(f(x) = C a^x + k\)). This shifts the graph up or down.
  • Horizontal Shifts: Changing the function by adding or subtracting a constant inside the exponent (e.g., \(f(x) = C a^{x-h}\)). This shifts the graph left or right.
  • Reflections: Multiplying by -1 to reflect the graph across the x-axis or y-axis.
  • Scaling: Multiplying by constants greater than 1 stretches the graph, while between 0 and 1 compresses it.
In our problem, the basic form \(f(x) = C \times a^x\) represents a function with no additional transformations. However, understanding these kinds of transformations is vital as they are frequently employed to adjust the base exponential function to more complex real-life applications.

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Most popular questions from this chapter

Mute swans were imported from Europe in the nineteenth century to grace ponds. Now there is concern that their population is growing too rapidly, edging out native species. Their population along the Atlantic coast has grown from 5800 in 1986 to 14,313 in 2002 . The increase is most acute in the mid-Atlantic region, but Massachusetts has also seen a jump, with 2939 mute swans counted in 2002 as compared with 585 in 1986 . a. Compare the growth factor in the mute swan population for the entire Atlantic coast with that for Massachusetts. b. Compare the average rate of change in the mute swan population for the entire Atlantic coast with that for Massachusetts. c. Construct both a linear and an exponential model for the mute swan population in Massachusetts since 1986 . d. Compare the projected populations of mute swans in Massachusetts by the year 2010 as predicted by your linear and exponential models.

Two cities each have a population of 1.2 million people. City A is growing by a factor of 1.15 every 10 years, while city \(\mathbf{B}\) is decaying by a factor of 0.85 every 10 years. a. Write an exponential function for each city's population \(P_{A}(t)\) and \(P_{B}(t)\) after \(t\) years. b. For each city's population function generate a table of values for \(x=0\) to \(x=50,\) using 10 -year intervals, then sketch a graph of each town's population on the same grid.

(Graphing program recommended.) On the same graph, sketch \(f(x)=3(1.5)^{x}, g(x)=-3(1.5)^{x},\) and \(h(x)=3(1.5)^{-x}\) a. Which graphs are mirror images of each other across the \(y\) -axis? b. Which graphs are mirror images of each other across the \(x\) -axis? c. Which graphs are mirror images of each other about the origin (i.e., you could translate one into the other by reflecting first about the \(y\) -axis, then about the \(x\) -axis)? d. What can you conclude about the graphs of the two functions \(f(x)=C a^{x}\) and \(g(x)=-C a^{x} ?\) e. What can you conclude about the graphs of the two functions \(f(x)=C a^{x}\) and \(g(x)=C a^{-x} ?\)

(Graphing program recommended.) Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react with nitrogen to produce radioactive carbon-14. Radioactive carbon-14 enters all living tissue through carbon dioxide (via plants). As long as a plant or animal is alive, carbon-14 is maintained in the organism at a constant level. Once the organism dies, however, carbon-14 decays exponentially into carbon-12. By comparing the amount of carbon- 14 to the amount of carbon-12, one can determine approximately how long ago the organism died. Willard Libby won a Nobel Prize for developing this technique for use in dating archaeological specimens. The half-life of carbon-14 is about 5730 years. In answering the following questions, assume that the initial quantity of carbon- 14 is 500 milligrams. a. Construct an exponential function that describes the relationship between \(A,\) the amount of carbon- 14 in milligrams, and \(t,\) the number of 5730 -year time periods. b. Generate a table of values and plot the function. Choose a reasonable set of values for the domain. Remember that the objects we are dating may be up to 50,000 years old. c. From your graph or table, estimate how many milligrams are left after 15,000 years and after 45,000 years. d. Now construct an exponential function that describes the relationship between \(A\) and \(T,\) where \(T\) is measured in years. What is the annual decay factor? The annual decay rate? e. Use your function in part (d) to calculate the number of milligrams that would be left after 15,000 years and after 45,000 years.

Estimate the doubling time using the rule of 70 when: a. \(P=2.1(1.0475)^{t}\), where \(t\) is in years b. \(Q=2.1(1.00475)^{T}\), where \(T\) is in years
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