Problem 23 - Precalculus: Mathematical for Calculus

Question

Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) (a) If \(f(x)=x^{3}-2 x+4,\) find \(f^{\prime}(a)\) (b) Find equations of the tangent lines to the graph of \(f\) at the points whose \(x\) -coordinates are \(0,1,\) and 2 (c) Graph \(f\) and the three tangent lines.

Step-by-Step Solution

Answer

(a) \(f'(a) = 3a^2 - 2\). (b) Tangent lines: \(y = -2x + 4\), \(y = x + 2\), \(y = 10x - 12\).

Step 1: Calculate the derivative of the function

To find \(f'(a)\), we first need the general derivative of \(f(x) = x^3 - 2x + 4\). We use the power rule for differentiation where the derivative of \(x^n\) is \(nx^{n-1}\). Thus, the derivative of \(f(x)\) is: \[ f'(x) = 3x^2 - 2\]

Step 2: Evaluate the derivative at x = a

To find \(f'(a)\), substitute \(a\) into the expression for \(f'(x)\): \[ f'(a) = 3a^2 - 2\] This gives the slope of the tangent line to the curve at any point \(x = a\).

Step 3: Determine equations of tangent lines at given x-coordinates

Given the x-coordinates 0, 1, and 2, we find the tangent lines using the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \(m = f'(x_1)\):For \(x = 0\): - \(f(0) = 4\) and \(f'(0) = -2\), so the equation is \(y - 4 = -2(x - 0)\) or \(y = -2x + 4\).For \(x = 1\): - \(f(1) = 3\) and \(f'(1) = 1\), so the equation is \(y - 3 = 1(x - 1)\) or \(y = x + 2\).For \(x = 2\): - \(f(2) = 8\) and \(f'(2) = 10\), so the equation is \(y - 8 = 10(x - 2)\) or \(y = 10x - 12\).

Step 4: Graph the function and tangent lines

When graphing, plot the function \(f(x) = x^3 - 2x + 4\). Then, on the same axes, plot the tangent lines: - \(y = -2x + 4\): tangent at \(x=0\). - \(y = x + 2\): tangent at \(x=1\). - \(y = 10x - 12\): tangent at \(x=2\).Ensure each tangent line just touches the curve at the respective point.

Key Concepts

Tangent Line

A tangent line is a straight line that touches a curve at exactly one point. At that point, the line has the same slope as the curve. The slope of a tangent line represents how steep the curve is at that particular point.
To find this slope, we use the derivative of the function at the given point. For a function like \(f(x) = x^3 - 2x + 4\), the derivative tells us the slope of the tangent line for any value of \(x\).
Here's a simple breakdown:

• Compute the derivative \(f'(x) = 3x^2 - 2\), which gives us a formula to find the slope at any point \(x = a\).
• For a specific point \(x = a\), plug \(a\) into \(f'(x)\) to find the slope \(f'(a)\).
• Use the slope and the point-slope form of a linear equation to find the equation of the tangent line.

In our instance, at various points like \(x=0\), \(x=1\), and \(x=2\), we compute the slope and equation of the tangent line using these steps. This way, each tangent line fits snugly against the curve at its assigned point.

Power Rule

The power rule is a basic rule in calculus used to find the derivative of functions in the form \(x^n\). It's a quick and efficient way to differentiate these types of functions. The rule states that if \(f(x) = x^n\), then the derivative \(f'(x) = nx^{n-1}\).
Using the power rule, you can easily calculate the derivative of each term in polynomial functions. For the function \(f(x) = x^3 - 2x + 4\), we apply the power rule to each term.

• For the term \(x^3\), the derivative is \(3x^2\).
• For the term \(-2x\), it becomes \(-2\), since the derivative of \(x\) is 1.
• The constant \(4\) has a derivative of zero because constants are flat on a graph.

Combining these results gives us \(f'(x) = 3x^2 - 2\), which is crucial for determining the slopes of tangent lines at any point on the graph.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to represent a mathematical equation visually. It helps us understand how the function behaves and can highlight features like intercepts, slopes, maxima, and minima.
To graph the function \(f(x) = x^3 - 2x + 4\), one must:

• Identify key features such as intercepts. The y-intercept is the point where the function crosses the y-axis. For this function, it occurs at \((0, 4)\).
• Consider the shape. A cubic function like this one typically has an "S" shape due to its highest power being \(x^3\).
• Plot a few points to guide the curve formation, then smooth out those points to reflect the continuous nature of the graph.
• Overlay tangent lines at points of interest \(x=0\), \(x=1\), \(x=2\) to visualize where these lines touch the curve, helping us see their slopes work as calculated.

Through graphing, you can observe the function's growth and direction changes, and the tangent lines represent how those changes play out at specific points.