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

Investment The value of a particular investment changes over time according to the function $$S(t)=5000 e^{0.1\left(e^{n m}\right)}$$ 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) $10,468$

Investment The value of a particular investment changes over time according to the function
$$S(t)=5000 e^{0.1\left(e^{n m}\right)}$$
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) $10,468$

Key Concepts

Exponential Growth

Exponential growth refers to a process where the change in a quantity over time is proportional to the current amount. This concept is frequently used in financial contexts to model how investments can increase in value when the growth rate causes the value to compound continuously.

Derivative of Exponential Functions

The derivative of an exponential function is key to understanding how rapidly a quantity is changing at any given moment. In exponential functions, the derivative often involves multiplying the original function by the derivative of the exponent, reflecting the nature of continuous change.

Chain Rule

The chain rule is a crucial tool in calculus for differentiating composite functions. It states that the derivative of an outer function, evaluated at an inner function, must be multiplied by the derivative of the inner function. This is particularly important when dealing with nested exponential functions in modeling practical phenomena.

Rate of Change in Investment Value

The rate of change of an investment's value is determined by differentiating its value function with respect to time. This derivative, often called the instantaneous rate of change, provides insight into how quickly the investment is appreciating or depreciating at a specific moment in time, which is critical for making informed financial decisions.

Transcript

00:01 Okay, so for number 44, we have this function, and we are asked to find the rate of change at which the value of the investment is changing after eight years.

00:11 So we want our t to equal eight.

00:15 This is a multiple choice question, but we need to find the rate, so that tells us we need to find the derivative.

00:24 But for this function here, we have this in the exponent.

00:28 So we are going to recall the property of differentiating exponential functions that says at e to the gt, so g of t is some function in the exponent.

00:49 If this is equal to y, then the derivative of y is equal to g prime of t, e to the gt, which tells this that we need to take the derivative of the exponent.

01:04 And this goes along with the regular exponential rules we've been working with so far.

01:09 So for this particular question, if we let g of t be equal to 0 .1e to the 0 .25t.

01:20 So that's what our exponent is.

01:21 So that's going to be g of t.

01:23 Then g prime of t, we can use regular exponentiation here, 0 .1e to the 0 .25.

01:30 So the whole exponential function times the derivative of the exponent which is 0 .25 and then we can simplify that to become 0 .025 e to the point 25 t so now we have our g prime of t and now we can actually find the entire derivative so s prime of t now is equal to the whole exponential function the whole entire exponential function which was 5 ,000 e to the point 1, e to the point 25t whole exponential function which we have here this kind of goes to this rule in blue that the derivative is equal to the whole exponential function right here times this derivative which is what's in the exponent so we're going to multiply this by this right here because that's our g prime of t times point 025e to the point 25 t so on the next page we're going to do some simplification so we're going to do this multiplication right here and we're going to combine the exponents so now we have that s prime of t is equal to we're going to combine the 5 ,000 and the 0 .25 and we have 125 e to the point 1e to the point 25 t times e to the 0 .25t and now we can combine these exponents just to make it look a little nicer e to the point one e.

03:24 0 .25t plus 0 .25t.

03:34 Let me rewrite this so it's just a little bit clearer give me give so you can tell what's in the exponent so if we rewrite this we should have 125e to the .1e to the .25t plus .25t...

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