Question

suppose that you are given the task of learning 100% of a block of knowledge. human nature is such that we retain only a percentage p of knowledge t weeks after we have learned it. the ebbinghaus - learning model assumes that p is given by p(t)=q+(100 - q)e^(-kt), where q is the percentage that we would never forget and k is a constant that depends on the knowledge learned. suppose that q = 45 and k = 0.1. complete parts (a) through (e) below.
(a) find the rate of change of p with respect to time t.
p(t)=
(e) interpret the meaning of the derivative. choose the correct answer below.
a. the derivative is the amount of time it takes to lose all knowledge that can be forgotten.
b. the derivative is the rate at which the retained knowledge changes per week.
c. the derivative represents the amount of knowledge retained forever.
d. the derivative is the rate at which exactly 10% of knowledge is lost.

Explanation:

Step1: Identify the function

We are given the function $P(t)=Q+(100 - Q)e^{-kt}$, with $Q = 45$ and $k=0.1$. So $P(t)=45+(100 - 45)e^{-0.1t}=45 + 55e^{-0.1t}$.

Step2: Differentiate using the chain - rule

The derivative of a constant is 0, and the derivative of $y = ae^{bx}$ with respect to $x$ is $y^\prime=abe^{bx}$. For $P(t)=45 + 55e^{-0.1t}$, the derivative $P^\prime(t)$:
The derivative of 45 is 0, and for $y = 55e^{-0.1t}$, using the formula $y^\prime=abe^{bx}$ where $a = 55$, $b=-0.1$, we get $P^\prime(t)=55\times(- 0.1)e^{-0.1t}=-5.5e^{-0.1t}$.

Step3: Interpret the derivative

The derivative of a function $y = f(t)$ with respect to $t$, $y^\prime=f^\prime(t)$ represents the rate of change of $y$ with respect to $t$. In the context of $P(t)$ which represents the percentage of knowledge retained over time $t$, $P^\prime(t)$ is the rate at which the retained knowledge changes per unit time.

Answer:

a) $P^\prime(t)=-5.5e^{-0.1t}$
b) B. The derivative is the rate at which the retained knowledge changes per unit time.