Problem 6 - Calculus and its applications

Question

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1 c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part (b). $$f(x)=x^{3}$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 3x^2 \), with slopes 12, 0, and 3 at \( x = -2, 0, 1 \).

Step 1: Graph the Function

To graph the function, plot the equation \( f(x) = x^3 \). This function is a cubic function which has a characteristic S-shaped curve. The curve passes through the origin (0,0) and is symmetric about the origin. It decreases in the second quadrant and increases in the first quadrant.

Step 2: Draw Tangent Lines

To draw tangent lines at the points where \( x = -2, 0, \) and \( 1 \), examine the graph and find these specific points. The tangent line at \( x = -2 \) will just touch the curve at \( x = -2 \), the line at \( x = 0 \) will be horizontal (as the derivative at \( x = 0 \) for cubic functions is 0), and the tangent at \( x = 1 \) touches the curve at this point.

Step 3: Derive the Function's Derivative

To find \( f'(x) \) using the limit definition of a derivative, start with:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]Substitute \( f(x) = x^3 \):\[ f(x+h) = (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \]Thus,\[ f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} \]This simplifies to:\[ = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} \]Factoring out \( h \):\[ = \lim_{h \to 0} (3x^2 + 3xh + h^2) \]Taking the limit as \( h \to 0 \):\[ f'(x) = 3x^2 \]

Step 4: Calculate Specific Derivatives

With \( f'(x) = 3x^2 \), determine the slopes at the specified points:- \( f'(-2) = 3(-2)^2 = 3 \times 4 = 12 \)- \( f'(0) = 3(0)^2 = 0 \)- \( f'(1) = 3(1)^2 = 3 \)These values represent the slopes of the tangents at \( x = -2, 0, \) and \( 1 \) respectively.

Key Concepts

Derivative

The derivative is a central concept in calculus, and it's all about how a function changes at a particular point. Think of it as the slope of a function at any given point. It's like asking, "How steep is this hill right here?". To find the derivative, we use the limit definition:

• The formula is: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
• This formula calculates the slope of the tangent line to the function at a specific point.
• Applying it to our function, \( f(x) = x^3 \), gives us \( f'(x) = 3x^2 \).

This derivative tells us the rate of change of the function \( f(x) = x^3 \). Simply put, it shows how the function's growth or decrease varies as the \(x\) value changes. Remember, a function's derivative provides insights into the behavior of the function—it highlights where the function is increasing or decreasing and helps us understand concavity.

Tangent Lines

Tangent lines are straight lines that touch a curve at exactly one point. They show the immediate slope of the curve at that exact point.

• At a point \( x \), the tangent line's slope is the same as the derivative at that point.
• When you draw a tangent on the function \( f(x) = x^3 \) at a certain \( x \) value, you're illustrating the function's rate of change there.
• For example, at \( x = 0 \), the tangent line is horizontal. This is because \( f'(0) = 0 \), indicating a zero slope—meaning no change at that point.

Tangent lines give a visual representation of how a function behaves locally. What's fascinating is how they can indicate whether the function is climbing up (positive slope) or sliding down (negative slope)!

Graphing Functions

Graphing a function helps visualize its behavior across different values. For a function like \( f(x) = x^3 \), which is a cubic function, the graph has a characteristic S-shape.

• The graph is symmetric about the origin, meaning it looks the same to the left and right of the \(y\)-axis.
• In the first quadrant, the function increases and curves upward, while in the second quadrant, it decreases.
• Graphing involves plotting points based on given \( x \) values and the corresponding \( f(x) \) values.

By graphing, you can easily interpret the nature of a function and understand how its slope varies. You can also spot key features such as intercepts, turning points, and symmetry. Thus, graphing is a crucial tool to explore functions and make predictions based on the visual data provided by the graph.