Theory and Examples


In Exercises 51–54,


b. Graph y = f(x) and y = f'(x) side by side using separate sets of coordinate axes, and answer the following questions.


y = x⁴/4

Verified step by step guidance

1. First, identify the function given: \( y = \frac{x^4}{4} \). This is a polynomial function, and we will need to find its derivative to graph \( y = f'(x) \).
2. To find the derivative \( f'(x) \), apply the power rule. The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). For \( y = \frac{x^4}{4} \), differentiate to get \( f'(x) = x^3 \).
3. Next, graph the original function \( y = \frac{x^4}{4} \). This is a quartic function, which is symmetric about the y-axis and has a minimum point at the origin (0,0). The graph will be a smooth curve opening upwards.
4. Now, graph the derivative \( y = f'(x) = x^3 \). This is a cubic function, which is an odd function and has a point of inflection at the origin (0,0). The graph will pass through the origin and have a shape that increases steeply as \( x \) moves away from zero.
5. Finally, compare the graphs of \( y = f(x) \) and \( y = f'(x) \). Notice how the derivative \( f'(x) \) represents the slope of the tangent to the curve \( y = f(x) \) at any point \( x \). Where \( f'(x) = 0 \), the original function \( f(x) \) has a horizontal tangent, indicating a local minimum or maximum.

Key Concepts

Function Graphing

Graphing a function involves plotting its values on a coordinate plane to visualize its behavior. For y = x⁴/4, the graph is a smooth curve that represents the function's output for each input x. Understanding the shape and key features, such as intercepts and symmetry, is crucial for analyzing the function and its derivative.

Derivative

The derivative of a function, denoted as f'(x), represents the rate of change or slope of the function at any given point. For y = x⁴/4, the derivative is f'(x) = x³, which helps in understanding how the function's slope changes. This concept is essential for analyzing the relationship between the function and its rate of change.

Relationship Between Function and Derivative

The graph of a function and its derivative provides insights into the function's behavior. The derivative indicates where the function is increasing or decreasing and identifies critical points like maxima, minima, and inflection points. By comparing y = f(x) and y = f'(x), one can understand how changes in the function's slope affect its overall shape.