Problem 13 - Applied Calculus

Question

Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph. $$ f(x)=e^{x}-10 x $$

Step-by-Step Solution

Answer

The graph has a local minimum at \( x = \ln(10) \). It decreases for \( x < \ln(10) \) and increases for \( x > \ln(10) \).

Step 1: Finding the Derivative

To analyze where the function is increasing or decreasing, we first find the derivative: \[ f'(x) = \frac{d}{dx}(e^x - 10x) = e^x - 10. \] This derivative will help us identify the critical points and intervals of monotonicity.

Step 2: Locating Critical Points

Critical points occur where the derivative is zero or undefined. Set\[ f'(x) = e^x - 10 = 0, \]which gives us\[ e^x = 10. \]Taking the natural logarithm of both sides results in\[ x = \ln(10). \]This is the only critical point of the function.

Step 3: Analyzing Monotonicity

We analyze the sign of the derivative on intervals divided by the critical point:1. For \( x < \ln(10) \), \( f'(x) = e^x - 10 < 0 \) since \( e^x < 10 \). The function is decreasing.2. For \( x > \ln(10) \), \( f'(x) = e^x - 10 > 0 \) since \( e^x > 10 \). The function is increasing.

Step 4: Understanding the Graph's Shape

The critical point at \( x = \ln(10) \) is a local minimum since the function changes from decreasing to increasing. The function is decreasing for \( x < \ln(10) \) and increasing for \( x > \ln(10) \). These changes in direction confirm the function’s V-shaped graph with a turning point.

Key Concepts

Derivatives

Derivatives are a key concept in calculus, used to measure how a function changes as its input changes. Essentially, the derivative provides the slope of the function's graph at any given point. For the function \( f(x) = e^x - 10x \), we differentiate to find \( f'(x) = e^x - 10 \).
This derivative tells us how the function behaves around every point \( x \). Knowing the derivative is essential to understand how and where the graph increases or decreases.

• If \( f'(x) > 0 \), the function is increasing at \( x \).
• If \( f'(x) < 0 \), the function is decreasing at \( x \).

This information is foundational for identifying where the graph has peaks, troughs, or flat areas.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. They are significant because they can indicate places where the function changes direction or levels off.
For the function \( f(x) = e^x - 10x \), the critical point can be found by setting the derivative to zero: \( e^x - 10 = 0 \). Solving gives \( e^x = 10 \), leading to \( x = \ln(10) \) as the only critical point.
This point is important because it is where the function's behavior switches from decreasing to increasing, making \( x = \ln(10) \) a local minimum. Understanding critical points helps us predict and visualize the overall shape of the graph.

Monotonicity

Monotonicity describes the intervals where a function consistently increases or decreases. To determine monotonicity, we examine the sign of the derivative across different intervals divided by critical points.
In our example, \( f(x) = e^x - 10x \), we have:

• For \( x < \ln(10) \), \( f'(x) = e^x - 10 \) is negative, indicating the function decreases.
• For \( x > \ln(10) \), \( f'(x) = e^x - 10 \) is positive, indicating the function increases.

The function transitions from decreasing to increasing at its critical point, confirming \( x = \ln(10) \) is a local minimum. Understanding monotonicity gives us insight into the function's behavior without having to graph it.

Exponential Functions

Exponential functions, like \( e^x \), are powerful and have unique characteristics. They grow or decay at a constant rate, making them indispensable in various scientific fields.
For \( f(x) = e^x - 10x \), the exponential term \( e^x \) dominates when \( x \) is large. This leads to rapid growth beyond any linear component, such as \(-10x\).
In calculus, the behavior of exponential functions underlies many key principles. Properties like constant relative growth rate and derivative equivalence \( \frac{d}{dx}(e^x) = e^x \) are especially useful. In our function, \( e^x \) helps shape the curve and influences where the function increases or decreases. Understanding exponential functions helps us predict both short-term fluctuations and long-term trends.