Problem 10 - Applied Calculus

Question

(a) Let \(g(t)=(0.8)^{t}\). Use a graph to determine whether \(g^{\prime}(2)\) is positive, negative, or zero. (b) Use a small interval to estimate \(g^{\prime}(2)\).

Step-by-Step Solution

Answer

(a) Negative; (b) \(-0.38391\)

Step 1: Analyze the Function's Behavior

The given function is an exponential decay function because the base 0.8 is less than 1. Therefore, function values decrease as \(t\) increases.

Step 2: Graph Interpretation

By analyzing the graph of \(g(t)=(0.8)^t\), we observe that the function is strictly decreasing. As \(t\) increases past 2, the function values continue to decline, indicating a negative slope at \(t=2\). Thus, \(g'(2)\) is negative.

Step 3: Estimate the Derivative using a Small Interval

Select a small interval around \(t=2\), such as \([2, 2.01]\). Calculate \(g(2)=(0.8)^{2}\) and \(g(2.01)=(0.8)^{2.01}\). The small interval approximation for \(g'(2)\) is given by \(\frac{g(2.01) - g(2)}{2.01 - 2}\).

Step 4: Calculate Function Values

Calculate \((0.8)^2 = 0.64\) and \((0.8)^{2.01} \approx 0.6361609\).

Step 5: Calculate the Derivative Approximation

Using the formula from Step 3, compute the derivative: \[ \frac{0.6361609 - 0.64}{0.01} = \frac{-0.0038391}{0.01} = -0.38391. \] This indicates \(g'(2)\) is approximately \(-0.38391\), confirming a negative slope.

Key Concepts

Derivative Estimation

In order to estimate the derivative of a function at a specific point, we choose points very close to it and calculate the rate of change between them. This technique is utilized by selecting a small interval that includes the point of interest. For example, in this exercise, we are estimating the derivative of the function \( g(t) = (0.8)^t \) at \( t = 2 \).To do this, you select an interval such as \([2, 2.01]\), because it's close and good enough to estimate the rate of change between these two points. Compute the function values at both ends of the interval: \( g(2) = (0.8)^2 \) and \( g(2.01) = (0.8)^{2.01} \).
Thus, the derivative is approximated using the difference quotient:

• \( \frac{g(2.01) - g(2)}{2.01 - 2} \)

This method provides an approximation of \( g'(2) \), giving us insight into the behavior of the function at the specified point.

Graph Interpretation

When you interpret a graph of a function, you're observing the visual behavior and trends of that function. For the exponential decay function \( g(t) = (0.8)^t \), the graph will display an overall downward trend as \( t \) increases. This is because an exponential function with a base less than 1 will decrease in value as the variable \( t \) gets larger.By looking at the graph around \( t = 2 \), you'll see that the slope is moving downward, confirming that the derivative \( g'(2) \) is negative. The graph helps you visualize abstract concepts by providing a real-world depiction of changes and patterns in the function. Always remember:

• If the curve is going down, the slope of the tangent line is negative.
• Graphs make it easy to identify trends and derive interpretations about the function's behavior.

Decreasing Function

A decreasing function, such as \( g(t) = (0.8)^t \), has a function value that becomes smaller as \( t \) increases. This is precisely the signature of an exponential decay function which occurs when the base of the exponent is less than 1.To technically state that a function is decreasing:

• The derivative \( g'(t) \) is negative wherever the function is decreasing.
  
• This implies that as you move from left to right on the graph, the output value of the function decreases.

For \( g(t) = (0.8)^t \), the function continues to decrease past \( t = 2 \), clearly showing a negative slope at \( t = 2 \), thus confirming a negative derivative.

Slope Calculation

Slope calculation involves determining how steep a line is. When assessing functions, the slope at a particular point translates into the derivative at that point. For instance, to find the slope of \( g(t) = (0.8)^t \) at \( t = 2 \), you evaluate how the function changes as \( t \) changes.Using the previously found small interval \([2, 2.01]\), compute:

• Function at \( t = 2 \) yielding \( g(2) = 0.64 \).
• Function at \( t = 2.01 \) yielding approximately \( g(2.01) \approx 0.6361609 \).

Then, the slope is calculated by the change in function values over the change in \( t \):

• \( \frac{0.6361609 - 0.64}{0.01} \approx -0.38391 \)

This negative value confirms a decreasing slope, thus confirming the function decreases at \( t = 2 \). Slope calculations are integral for understanding the rate at which values increase or decrease, helping in analyzing the behavior of functions.