Problem 14 - Calculus

Question

Sketch the graph of \(f(x),\) and use this graph to sketch the graph of \(f^{\prime}(x)\). $$f(x)=5 x$$

Step-by-Step Solution

Answer

Both graphs are straight lines: \(f(x) = 5x\) through the origin, \(f'(x) = 5\) horizontal at y=5.

Step 1: Understand the Function

The given function is linear, \(f(x) = 5x\). A linear function is of the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, \(m = 5\) and \(b = 0\), indicating the line is steep and passes through the origin (0,0).

Step 2: Sketch the Graph of \(f(x)\)

To sketch \(f(x) = 5x\), draw a straight line that passes through the origin (0,0) and has a slope of 5. As this is a linear function, for every 1 unit increase in \(x\), \(f(x)\) increases by 5 units. The graph will be a straight line tilting upwards to the right.

Step 3: Find the Derivative \(f'(x)\)

Since \(f(x) = 5x\), we apply the power rule for derivatives, where \(f'(x) = 5\times x^{0} = 5\). Thus, the derivative \(f'(x)\) is a constant function, \(f'(x) = 5\).

Step 4: Sketch the Graph of \(f'(x)\)

The derivative \(f'(x) = 5\) is a horizontal line because it is a constant function. This line will be parallel to the x-axis and will intersect the y-axis at y = 5.

Key Concepts

Linear Function

A linear function is one of the simplest kinds of functions in mathematics. It is represented by the equation \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The crucial features of a linear function include:

• The slope \(m\), which determines the steepness of the line and the direction of its tilt. A positive slope means the line tilts upwards as you move from left to right.
• The y-intercept \(b\), indicating where the line intersects the y-axis.

In our function \(f(x) = 5x\), \(m = 5\) and \(b = 0\). This means the line is fairly steep and passes through the origin (0, 0). Recognizing these characteristics helps us quickly sketch and differentiate these functions.

Graph Sketching

Graph sketching is an essential skill in analyzing mathematical functions. For a linear function like \(f(x) = 5x\), sketching is straightforward:

• Start at the y-intercept (0,0) because \(b=0\).
• Use the slope \(m=5\). From the y-intercept, move up 5 units vertically and 1 unit to the right horizontally.

These steps will give you another point on the line, aiding in drawing the straight line accurately.
For sketching derivatives, understand that if \(f(x)\) is linear, \(f'(x)\) is a constant. Here, \(f'(x) = 5\) results in a horizontal line at \(y = 5\). This visual aspect can be very informative in understanding the behavior of functions.

Power Rule

The power rule is a key concept in differentiation, which helps in finding the derivative of functions of the form \(x^n\). The rule states that the derivative of \(x^n\) is \(nx^{n-1}\). For a linear function, consider \(f(x) = 5x = 5x^1\). Applying the power rule:

• The exponent 1 multiplied by the coefficient 5 gives 5.
• The exponent decreases by one to become 0, resulting in \(x^0 = 1\).

Thus, the derivative \(f'(x) = 5\times1 = 5\) always holds true. This shows how a linear function simplifies under differentiation, providing foundational insight for more complex calculus tasks.