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Mathematics LibreTexts

4.E: Exponential and Logarithmic Functions (Exercises)

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    17849
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    4.1: Exponential Functions

    When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions, which model this kind of rapid growth.

    Verbal

    1) Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function.

    Answer:

    Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original.

    2) Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.

    3)The Oxford Dictionary defines the word nominal as a value that is “stated or expressed but not necessarily corresponding exactly to the real value.” Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.

    Answer:

    When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account. Thus, the annual percentage rate does not necessarily correspond to the real interest earned, which is the very definition of nominal.

    Algebraic

    For the following exercises, identify whether the statement represents an exponential function. Explain.

    4) The average annual population increase of a pack of wolves is \(25\).

    5) A population of bacteria decreases by a factor of \(\frac{1}{8}\) every \(24\) hours.

    Answer:

    exponential; the population decreases by a proportional rate.

    6) The value of a coin collection has increased by \(3.25\%\)annually over the last \(20\) years.

    7) For each training session, a personal trainer charges his clients \(\$5\) less than the previous training session.

    Answer:

    not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function.

    8) The height of a projectile at time \(t\) is represented by the function \(h(t)= -4.9t^2 + 18t + 40\)

    For the following exercises, consider this scenario: For each year

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