Problem 118 - Calculus Volume 1

Question

For the following exercises, find the equation of the tangent line \(T(x)\) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. $$[\mathrm{T}] \mathrm{y}=3 x^{2}+4 x+1\( at \)(0,1)$$

Step-by-Step Solution

Answer

The tangent line is \( T(x) = 4x + 1 \).

Step 1: Find the derivative

The first step in finding the equation of the tangent line is to find the derivative of the given function, which represents the slope of the tangent line. The function is \( y = 3x^2 + 4x + 1 \). Differentiate the function with respect to \( x \):\[ \frac{dy}{dx} = 6x + 4. \]

Step 2: Evaluate the derivative at the given point

Now, we need to find the slope of the tangent line at the specific point provided, which is \((0,1)\). Substitute \(x = 0\) into the derivative:\[ \frac{dy}{dx} = 6(0) + 4 = 4. \] Thus, the slope of the tangent line at \(x = 0\) is 4.

Step 3: Use the point-slope form to find the equation of the tangent line

With the slope \(m = 4\) and the point \((0,1)\), we can use the point-slope form of the equation of a line to find the equation of the tangent line. The point-slope form is given by:\[ T(x) = m(x - x_1) + y_1,\]where \((x_1, y_1)\) is the point on the line. Substitute \(m = 4\), \(x_1 = 0\), and \(y_1 = 1\):\[ T(x) = 4(x - 0) + 1 = 4x + 1.\]Thus, the equation of the tangent line is \( T(x) = 4x + 1 \).

Step 4: Graph the function and the tangent line

Use a graphing calculator to plot the original function \( y = 3x^2 + 4x + 1 \) and the tangent line \( T(x) = 4x + 1 \). Ensure that both graphs intersect at the point \((0, 1)\). By graphing, you can visually confirm that the tangent line touches the curve at this point without crossing it.

Key Concepts

Derivative

The derivative of a function is a core concept in calculus that helps us understand how a function is changing at a specific point. It represents the instantaneous rate of change or the slope of the tangent line at any given point on a curve. To find the derivative, we apply differentiation rules such as the power rule, product rule, or chain rule, depending on the function’s composition.
For the function in our exercise, which is \( y = 3x^2 + 4x + 1 \), we use the power rule to differentiate each term. The power rule states that for any term of the form \( ax^n \), the derivative is \( n \cdot ax^{n-1} \).

• Differentiating \( 3x^2 \), we get \( 6x \).
• Differentiating \( 4x \), we get \( 4 \).
• The derivative of a constant, like \( 1 \), is \( 0 \).

This process yields the derivative \( \frac{dy}{dx} = 6x + 4 \), which we then evaluate to find the slope of the tangent line at a given point.

Point-Slope Form

The point-slope form is a helpful equation used to find the equation of a line when you know the slope and a point on the line. It is particularly useful when dealing with tangent lines in calculus.
The formula is given as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line. This form quickly gives us the equation without needing to first solve for the y-intercept, a step required in the slope-intercept form.
For our exercise, once the derivative was calculated and evaluated at \( x = 0 \), we found that \( m = 4 \). Using this slope and the point \((0,1)\), we substituted into the point-slope formula:
\[ y - 1 = 4(x - 0) \]
Simplifying this, we get the tangent line equation \( T(x) = 4x + 1 \). This representation connects the concept of the derivative directly to finding tangible line equations.

Slope of Tangent Line

Knowing the slope of a tangent line is crucial because it tells us how steep the line is at a specific point on a curve. The slope is essentially a snapshot of the behavior of a function at that bar point, reflecting an immediate change.
In our provided example, once we had the derivative \( \frac{dy}{dx} = 6x + 4 \), we evaluated it at the point \( x = 0 \) to find \( 6(0) + 4 = 4 \). Therefore, the slope of the tangent line at \( x = 0 \) is \( 4 \).

• This indicates that the tangent line rises 4 units vertically for every 1 unit it moves horizontally.
• In the context of the parabolic curve, it shows the direction and steepness of the tangent at exactly one spot.

This understanding helps predict function behavior and supports further exploration of calculus concepts like concavity and inflection points.

Graphing Calculator

Graphing calculators are powerful tools for visualizing equations and functions, making abstract concepts more concrete and easier to understand. They allow you to input equations to see their graph, assisting in checking the results of your calculations like finding tangent lines.
When you graph the original function \( y = 3x^2 + 4x + 1 \) and the tangent line equation \( T(x) = 4x + 1 \) on a graphing calculator, you can visually confirm the outcome of previous calculations. Ensuring both graphs intersect neatly at the point \( (0, 1) \) reveals that our tangent line approximation is accurate.

• The visual confirmation aids comprehension of how the slope affects the tangent’s positioning.
• Helps verify if the tangent just touches the curve at precisely one point.
• Provides insight into how changes in function coefficients modify their respective graphs.

Using a graphing calculator bridges gap between theoretical calculations and tangible graph interpretation, enriching learning experiences.