Problem 69 - Calculus and its applications

Question

The rate of memorizing information initially increases. Eventually, however, a maximum rate is reached, after which it begins to decrease. Suppose an experiment finds that the rate of memorizing is given by $$M^{\prime}(t)=-0.009 t^{2}+0.2 t$$ where \(M^{\prime}(t)\) is the memory rate, in words per minute. How many words are memorized in the first \(10 \mathrm{~min}\) \((\) from \(t=0\) to \(t=10) ?\)

Step-by-Step Solution

Answer

7 words are memorized in the first 10 minutes.

Step 1: Identify the Function to Integrate

We need to find the total number of words memorized in the first 10 minutes, which means we need to integrate the memory rate function from 0 to 10 minutes. The memory rate function given is \( M'(t) = -0.009t^2 + 0.2t \).

Step 2: Setup the Integral

Set up the definite integral for the memory rate function from \( t = 0 \) to \( t = 10 \). The integral is: \[ \int_{0}^{10} (-0.009t^2 + 0.2t) \, dt \]

Step 3: Integrate the Function

Integrate the function \(-0.009t^2 + 0.2t\). The anti-derivative is: \(-0.009 \frac{t^3}{3} + 0.2 \frac{t^2}{2}\). Simplifying, this becomes: \(-0.003t^3 + 0.1t^2\).

Step 4: Evaluate the Definite Integral

Plug in the limits of integration. \[ \left( -0.003(10)^3 + 0.1(10)^2 \right) - \left( -0.003(0)^3 + 0.1(0)^2 \right) \] Compute the value: \[ -0.003(1000) + 0.1(100) - (0) = -3 + 10 = 7 \]

Step 5: Interpret the Result

The definite integral from 0 to 10 gives the total number of words memorized in the first 10 minutes. Thus, 7 words are memorized in that time frame.

Key Concepts

Memory Rate Function

The memory rate function is essential to understand how quickly information is learned over time. In the given exercise, it is expressed mathematically as \( M'(t) = -0.009t^2 + 0.2t \). This function reflects the rate of memorization, or how many words per minute a person can memorize at any given point in time. - The components of the function, \(-0.009t^2\) and \(0.2t\), have specific roles: - The \(-0.009t^2\) term indicates that as time increases, the memorization rate will eventually start to decrease, as denoted by the negative coefficient. - The \(0.2t\) term contributes to the initial increase in the memorization rate.The curve represented by the memory rate function starts to climb initially, rises to a peak, and then descends. It represents a realistic model where a person becomes proficient at memorizing up to a certain point, but then cognitive fatigue or the natural decline in rate with time reduces the efficiency.

Definite Integral

The concept of the definite integral is used to find the total amount accumulated over an interval. In our exercise, we seek the total words memorized over the first ten minutes by calculating the area under the memorization rate curve between \(t = 0\) and \(t = 10\). - The integral expression, \( \int_{0}^{10} (-0.009t^2 + 0.2t) \, dt \), sets the boundaries from zero to ten minutes. - Evaluating a definite integral involves finding the antiderivative first, which, when evaluated at the upper and lower bounds, gives us the net accumulation of words memorized.This integration is akin to adding up small slices of knowledge gained every minute from \(t = 0\) to \(t = 10\). This cumulative process effectively tells us the total contribution of our memory rate over time.

Antiderivative

Finding the antiderivative is crucial to solving the definite integral of a function. The antiderivative reverses the process of differentiation and provides us the original function from a rate of change.- For the memory rate function \(M'(t) = -0.009t^2 + 0.2t\), the antiderivative becomes \(-0.003t^3 + 0.1t^2\). - The antiderivative is found by applying simple power rules: increasing the exponent by one and dividing by the new exponent. - Each term in the derivative follows this process separately. Computing the antiderivative helps in moving from the rate of memorization to the total memorization. It enables us to evaluate how much total learning has occurred over a specified time period. This step is vital to transform from a rate (words per minute) into an accumulated total (total words).