Problem 60 - Applied Calculus

Question

Tablet Computers When tablet computers were first introduced, sales grew at the rate of \(21 x e^{0.1 x}\) million per year. Find the cumulative sales over the first five years.

Step-by-Step Solution

Answer

Cumulative sales over 5 years is approximately 1573 million units.

Step 1: Define the problem

We need to find the cumulative sales of tablet computers over the first 5 years given a rate of sales growth function. This involves calculating the integral of the rate function from 0 to 5.

Step 2: Write the integral

The sales growth rate function is given by \(21x e^{0.1x}\). To find the cumulative sales over the first 5 years, set up the integral: \[\int_0^5 21x e^{0.1x} \, dx\].

Step 3: Choose the integration method

The function \(21x e^{0.1x}\) requires integration by parts. Choose \(u = 21x\) and \(dv = e^{0.1x} \, dx\). This means \(du = 21 \, dx\) and \(v = \frac{1}{0.1}e^{0.1x}\).

Step 4: Apply integration by parts formula

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), calculate:\[ \int 21x e^{0.1x} \, dx = 21x \cdot \frac{1}{0.1} e^{0.1x} - \int \frac{1}{0.1} e^{0.1x} \cdot 21 \, dx \].

Step 5: Simplify and solve the integrals

First simplify:\[ 210x e^{0.1x} - 210 \int e^{0.1x} \, dx \].Calculate the integral:\[ 210 \cdot \frac{1}{0.1} e^{0.1x} = 2100 e^{0.1x} \].Thus the expression becomes:\[ 210x e^{0.1x} - 2100 e^{0.1x} + C \].

Step 6: Evaluate the definite integral

Evaluate from 0 to 5:\[\left[ 210(5) e^{0.5} - 2100 e^{0.5} \right] - \left[ 210(0) e^{0} - 2100 e^{0} \right]\].Simplify to:\[210(5)e^{0.5} - 2100e^{0.5} - 2100(1)\].Complete the calculation to find the total sales.

Step 7: Calculate the numerical value

Compute the numerical approximations for the expression:\[ 1050 e^{0.5} - 2100 e^{0.5} - 2100 \approx (1050 - 2100)e^{0.5} - 2100\].With \(e^{0.5} \approx 1.6487\), calculate to get the cumulative sales approximately.

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. It's a powerful tool to analyze and understand dynamic systems. Calculus is broadly divided into two branches:

• Differential Calculus: Focuses on the concept of the derivative, which measures how a function changes as its input changes. It's about finding the rate at which one quantity changes concerning another.
• Integral Calculus: Concerns the accumulation of quantities, such as areas under a curve or the total volume under a surface. This is achieved through the process of integration.

In our problem, integral calculus allows us to accumulate the sales over a period when given a rate of sales growth. We use techniques such as integration by parts to solve integrals that are not straightforward. This method applies the formula:\[ \int u \, dv = uv - \int v \, du \]which helps in integrating products of functions.

Definite Integral

A definite integral represents the accumulation of a quantity, such as area, sales, or volume, over a distinct interval on the x-axis. The notation for a definite integral is:\[ \int_{a}^{b} f(x) \, dx \]which means we integrate the function \(f(x)\) from \(a\) to \(b\), where these values form the limits of integration.
The result is a number that represents the total accumulation from \(a\) to \(b\). In the context of our problem, the definite integral helps us calculate the total tablet sales over the first five years by integrating the sales growth rate function \(21x e^{0.1x}\) from \(0\) to \(5\).
Calculating this involves determining the antiderivative (a function \(F(x)\) whose derivative is the original function \(f(x)\)) and evaluating it at points \(a\) and \(b\). In the exercise, this was achieved using:\[ \left[ F(b) - F(a) \right] \]to find the total accumulation.

Exponential Growth

Exponential growth occurs when the growth rate of a quantity is proportional to its current value, leading to its rapid increase over time. The mathematical representation involves exponential functions, typically noted as \(e^{kx}\), where:

• \(e\) is the base of natural logarithms, approximately 2.718.
• \(k\) is a positive constant that influences the rate of growth.
• \(x\) represents time or another variable of growth.

In our textbook problem, the growth rate of tablet sales involves an exponential factor \(e^{0.1x}\).
This implies that sales increase exponentially over time at a rate controlled by the constant 0.1. By integrating a sales growth rate function involving \(x e^{0.1x}\), we capture the essence of exponential growth over a specific period, thus allowing us to calculate cumulative sales efficiently.