The value of a particular investment changes over time...

The value of a particular investment changes over time according to the function $$S(t)=5000 e^{0.1\left(e^{0.25}\right)}$$ where $S(t)$ is the value after $t$ years. Calculate the rate at which the value of the investment is changing after 8 years. (Choose one of the following.) Source: Society of Actuaries. a. 618 b. 1934 c. 2011 d. 7735 e. 10,468

The value of a particular investment changes over time according to the function

$$S(t)=5000 e^{0.1\left(e^{0.25}\right)}$$

where $S(t)$ is the value after $t$ years. Calculate the rate at which the value of the investment is changing after 8 years. (Choose one of the following.) Source: Society of Actuaries.
a. 618
b. 1934
c. 2011
d. 7735
e. 10,468

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Key Concepts

Exponential Growth Functions

Exponential growth functions model situations in which the rate of change of a quantity is proportional to the current value of the quantity. This type of function is frequently encountered in finance, populations, and natural phenomena, and is generally written in the form f(t) = A·e^(kt), where A is an initial constant and k is a constant that affects the growth rate.

Differentiation of Exponential Functions

The differentiation of exponential functions relies on the fact that the derivative of e^(g(t)) is e^(g(t)) multiplied by the derivative of the exponent g(t). This rule, often written as d/dt[e^(g(t))] = e^(g(t))·g'(t), is crucial for calculating instantaneous rates of change in models described by exponential functions.

Chain Rule

The chain rule is a central technique in calculus used to differentiate composite functions, where one function is nested inside another. When you have a function defined as the composition of two or more functions, the chain rule states that you multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function. This rule is especially important when the exponent in an exponential function itself is dependent on another variable.

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