Problem 25 - Precalculus Mathematics for Calculus

Question

(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) \( f'(x) = 3x^2-2 \). (b) Tangent lines: \( y=-2x+4, y=x+1, y=10x-16 \). (c) Graph the function and lines.

Step 1: Find the Derivative

To find \( f'(x) \), we need to differentiate \( f(x) = x^3 - 2x + 4 \). Using the power rule, the derivative of \( x^3 \) is \( 3x^2 \), the derivative of \( -2x \) is \( -2 \), and the constant \( 4 \) becomes \( 0 \). Thus, \( f'(x) = 3x^2 - 2 \).

Step 2: Evaluate the Derivative at Given Points

We need to evaluate \( f'(x) = 3x^2 - 2 \) at \( x = 0, 1, \) and \( 2 \).- At \( x = 0 \), \( f'(0) = 3(0)^2 - 2 = -2 \).- At \( x = 1 \), \( f'(1) = 3(1)^2 - 2 = 1 \).- At \( x = 2 \), \( f'(2) = 3(2)^2 - 2 = 10 \).

Step 3: Find Tangent Lines' Equations

The equation of the tangent line at a point \( (a, f(a)) \) is given by \( y = f'(a)(x - a) + f(a) \).- For \( x = 0 \): \( y = -2(x - 0) + f(0) = -2x + 4 \) as \( f(0) = 4 \).- For \( x = 1 \): \( y = 1(x - 1) + f(1) = x + 1 \) as \( f(1) = 3 \).- For \( x = 2 \): \( y = 10(x - 2) + f(2) = 10x - 16 \) as \( f(2) = 8 \).

Step 4: Sketch the Graph

To sketch the graph of \( f(x) = x^3 - 2x + 4 \) along with the tangent lines, plot the function and draw the lines:- The curve \( x^3 - 2x + 4 \) is a standard cubic curve shape, going through the origin.- The tangent line \( y = -2x + 4 \) will just intersect the curve at \( x = 0 \).- The tangent line \( y = x + 1 \) will intersect the curve at \( x = 1 \).- The tangent line \( y = 10x - 16 \) will intersect the curve at \( x = 2 \).

Key Concepts

Tangent Lines in Calculus

A tangent line is a straight line that just touches a curve at a single point. This means it has the same slope as the curve at that point. When you're working with functions, finding the tangent line is crucial to understanding how the function behaves near that specific point.

To find the tangent line to a graph of a function at a certain point, you first need its slope. This slope is given by the derivative of the function evaluated at the point of interest. Once you have the slope, the equation of the tangent line can be written as:

• \( y = f'(a)(x - a) + f(a) \)

Here, \( a \) is the x-coordinate of the point and \( f'(a) \) is the slope of the tangent line at that point. In simpler terms, you're using the derivative to figure out how steep the tangent line should be.

Once you determine the slope, you can plug back into the equation to find the exact tangent line at any given point. In exercises, you would often be required to calculate for different values like 0, 1, or 2, which will give you different tangent lines at those specific points.

Power Rule in Calculus: Differentiation Made Simple

Differentiation is a way to find out how a function changes at any given point. One of the simplest and most frequently used rules for finding derivatives is the power rule.

The power rule states that if you have a function of the form \( f(x) = x^n \), the derivative \( f'(x) \) is \( nx^{n-1} \). This makes it easy to differentiate polynomials, where each term is simplified using this rule.

In the exercise example, for the function \( f(x) = x^3 - 2x + 4 \), we applied the power rule:

• For \( x^3 \), the derivative is \( 3x^2 \).
• For \( -2x \), it becomes \( -2 \), as \( n = 1 \) means \( 1x^{1-1} = 2x^0 = 2 \).
• The constant 4 disappears since its derivative is \( 0 \).

Thus, by combining derivatives of individual terms, we get \( f'(x) = 3x^2 - 2 \), which can then be used for finding slopes of tangent lines or for analyzing the function further.

Graphing Cubic Functions

Graphing functions can be an insightful way to understand their behavior. Cubic functions, in particular, have their unique shape characteristics that can tell a lot about how the function behaves.

The function given in the exercise, \( f(x) = x^3 - 2x + 4 \), is a cubic polynomial. It inherently has a single bend, presenting an S-shape or inverted S-shape depending on its equation. Here's how you might approach graphing a cubic function like this:

First, find key points on the graph, such as:

• The y-intercept, which is the constant term, here 4.
• Points where the curve cuts through the x-axis, if it does. These are the roots of the equation, and you can find them by solving \( f(x) = 0 \).

Once you plot these critical points, map the approximate curvature guided by the shape of cubic functions. Adding tangent lines can give additional insight, as they capture how steep or flat the curve is at specific points like \( x = 0, 1, \) and \( 2 \). These tangent lines intersect the curve precisely at these x-values. When sketching the graph with tangent lines, each tangent line hints at the curve's slope at the point of contact, providing further visual understanding of the function's gradient at those points.