Problem 49 - Calculus: Early Transcendentals

Question

(a) Evaluate the function \( f(x) = x^2 - (2^x/1000) \) for \( x \) = 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of $$ \lim_{x \to 0}\left( x^2 - \frac{2^x}{1000} \right) $$ (b) Evaluate \( f(x) \) for \( x \) = 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.

Step-by-Step Solution

Answer

The limit of the function as \( x \to 0 \) is approximately \(-0.001\).

Step 1: Evaluating the Function for Initial Values

First, we evaluate the function \( f(x) = x^2 - \frac{2^x}{1000} \) at the given values of \( x \). For each value, we simply substitute \( x \) into the function and solve:1. For \( x = 1 \),\[ f(1) = 1^2 - \frac{2^1}{1000} = 1 - 0.002 = 0.998 \]2. For \( x = 0.8 \),\[ f(0.8) = 0.8^2 - \frac{2^{0.8}}{1000} \approx 0.64 - 0.00174 = 0.63826 \]3. For \( x = 0.6 \),\[ f(0.6) = 0.6^2 - \frac{2^{0.6}}{1000} \approx 0.36 - 0.00152 = 0.35848 \]4. For \( x = 0.4 \),\[ f(0.4) = 0.4^2 - \frac{2^{0.4}}{1000} \approx 0.16 - 0.00132 = 0.15868 \]5. For \( x = 0.2 \),\[ f(0.2) = 0.2^2 - \frac{2^{0.2}}{1000} \approx 0.04 - 0.00115 = 0.03885 \]6. For \( x = 0.1 \),\[ f(0.1) = 0.1^2 - \frac{2^{0.1}}{1000} \approx 0.01 - 0.00107 = 0.00893 \]7. For \( x = 0.05 \),\[ f(0.05) = 0.05^2 - \frac{2^{0.05}}{1000} \approx 0.0025 - 0.00103 = 0.00147 \]

Step 2: Guessing the Limit from Initial Values

From the calculated values, we can observe that as \( x \) approaches zero, \( f(x) \) is getting closer to zero since each term \( x^2 - \frac{2^x}{1000} \) becomes smaller. Therefore, we can guess that\[ \lim_{x \to 0} \left( x^2 - \frac{2^x}{1000} \right) \approx 0 \]

Step 3: Evaluating the Function for Smaller Values

Now, evaluate the function \( f(x) \) for smaller values of \( x \):1. For \( x = 0.04 \),\[ f(0.04) = 0.04^2 - \frac{2^{0.04}}{1000} \approx 0.0016 - 0.00102 = 0.00058 \]2. For \( x = 0.02 \),\[ f(0.02) = 0.02^2 - \frac{2^{0.02}}{1000} \approx 0.0004 - 0.00101 = -0.00061 \]3. For \( x = 0.01 \),\[ f(0.01) = 0.01^2 - \frac{2^{0.01}}{1000} \approx 0.0001 - 0.00100 = -0.00090 \]4. For \( x = 0.005 \),\[ f(0.005) = 0.005^2 - \frac{2^{0.005}}{1000} \approx 0.000025 - 0.00100 = -0.000975 \]5. For \( x = 0.003 \),\[ f(0.003) = 0.003^2 - \frac{2^{0.003}}{1000} \approx 0.000009 - 0.00100 = -0.000991 \]6. For \( x = 0.001 \),\[ f(0.001) = 0.001^2 - \frac{2^{0.001}}{1000} \approx 0.000001 - 0.00100 = -0.000999 \]

Step 4: Observing the Behavior with Smaller Values

The evaluations for smaller \( x \) show that the values of \( f(x) \) quickly approach \(-0.001\) as \( x \) gets close to zero. Given the progressive negative increase, we guess that\[ \lim_{x \to 0} \left( x^2 - \frac{2^x}{1000} \right) \approx -0.001 \]

Key Concepts

Function Evaluation

In calculus, evaluating a function means determining the output of that function for specific input values of the variable. This exercise involves evaluating the function \( f(x) = x^2 - \frac{2^x}{1000} \) for several values of \( x \). To do this, we substitute each value of \( x \) into the function and simplify.

• When \( x = 1 \), substituting into the function gives \( f(1) = 1^2 - \frac{2^1}{1000} = 0.998 \).
• Repeat this process for \( x = 0.8 \), \( x = 0.6 \), and so on, to get the respective function values.

The purpose of evaluating the function at these points is to understand how the function behaves at different values, especially as \( x \) approaches zero.

Limit Estimation

Limit estimation involves predicting the behavior of a function as the input approaches a certain value, which is often zero in such problems. In this exercise, we look to estimate the limit \( \lim_{x \to 0} (x^2 - \frac{2^x}{1000}) \). The initial evaluations show that as \( x \) decreases, the output of the function becomes smaller and tends toward values like 0.00147 with smaller positive or even negative results for very small values of \( x \).

• For instance, with smaller \( x \) values such as 0.02, the function dips into negative values, suggesting convergence towards a particular negative constant.
• The objective is to observe this behavior and infer the limiting value as \( x \) inches closer to zero, creating an insight into the function's trend or tendency.

Behavior of Functions as x Approaches Zero

Understanding the behavior of functions as \( x \) approaches zero is a crucial part of calculus. When evaluating \( f(x) = x^2 - \frac{2^x}{1000} \) for descending values of \( x \), it becomes apparent that each term reacts differently. - The term \( x^2 \) shrinks significantly because the square of progressively smaller numbers quickly approaches zero.- On the other hand, \( \frac{2^x}{1000} \) remains quite stable, slightly above \( 0.001 \). Consequently, the subtraction \( x^2 - \frac{2^x}{1000} \) initially yields positive values but begins to decline, indicating a predominant influence of the second term as observed in values like \(-0.000999\) for \( x = 0.001 \).The key is to note how the function manages these competing terms close to \( x = 0 \), leading to an estimated limit where the function swings toward a negative constant value, \(-0.001\). This insight is significant when predicting long-term behavior or approximating values near specific points.