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5.3.1: Graphs and Properties of Exponential Growth and Decay Functions (Exercises)

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    39446
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    SECTION 5.3 PROBLEM SET: GRAPHS AND PROPERTIES OF EXPONENTIAL GROWTH AND DECAY FUNCTIONS

    In questions 1-4, let \(t\) = time in years and \(y\) = the value at time \(t\) or \(y\) = the size of the population at time \(t\). The domain is the set of non-negative values for \(t\); \(t ≥ 0\), because \(y\) represents a physical quantity and negative values for time may not make sense. For each question:

    1. Write the formula for the function in the form \(y = ab^t\)
    2. Sketch the graph of the function and mark the coordinates of the y-intercept.
    1. A house was purchased for $350,000 in the year 2010. The value has been increasing at the rate of 2% per year.
    1. A population of a certain species of bird in a state park has 300 birds. The population is decreasing at the rate of 7% year.
    1. A lab buys equipment $50,000. Its value depreciates over time. The value decreases at the rate of 6% annually.
    1. A population of bats in a cave has 200 bats. The population is increasing at the rate of 5% annually.

    In questions 5-8, let \(t\) = time in years and \(y\) = the value at time \(t\) or \(y\) = the size of the population at time \(t\). The domain is the set of non-negative values for \(t\); \(t ≥ 0\), because \(y\) represents a physical quantity and negative values for time may not make sense. For each question:

    1. Write the formula for the function in the form \(y = ae^{kt}\)
    2. Sketch the graph of the function and mark the coordinates of the y-intercept.
    1. A population of 400 microbes increases at the continuous growth rate of 26% per day.
    1. The price of a machine needed by a production factory is $28,000. Due to inflation the price of the machine is increasing at the continuous rate of 3.5% per year.
    1. A population of an endangered species consists of 4000 animals of that species. The population is decreasing at the continuous rate of 12% per year.
    1. A business buys a computer system for $12000. The value of the system is depreciating and decreases at the continuous rate of 20% per year.

    For questions 9-12

    1. Sketch a graph of exponential function.
    2. List the coordinates of the y intercept.
    3. State the equation of any asymptotes and state the whether the function approaches the asymptote as x →∞ or as x→ −∞ .
    4. State the domain and range.
    1. \(y=10\left(1.5^{x}\right)\)
    1. \(y=10\left(e^{1.2 x}\right)\)
    1. \(y=32\left(0.75^{x}\right)\)
    1. \(y=200\left(e^{-0.5 x}\right)\)

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