Problem 45 - Calculus for Biology and Medicine

Question

In Problems , find the tangent line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=2 x^{4}-5 x, \text { at } x=1 $$

Step-by-Step Solution

Answer

The tangent line in standard form is \( 3x - y = 6 \).

Step 1: Find the derivative

To find the derivative, which gives the slope of the tangent line, we differentiate the function. The function is given by \( y = f(x) = 2x^4 - 5x \). The derivative \( f'(x) \) is calculated as follows: \[ f'(x) = \frac{d}{dx}(2x^4 - 5x) = 8x^3 - 5. \]

Step 2: Evaluate the derivative at the point

Now, we need to find the slope of the tangent line at \( x = 1 \). Substitute \( x = 1 \) into the derivative: \[ f'(1) = 8(1)^3 - 5 = 8 - 5 = 3. \]The slope of the tangent line at \( x = 1 \) is 3.

Step 3: Find the y-coordinate of the point

Substitute \( x = 1 \) into the original function to find the y-coordinate of the point:\[ y = 2(1)^4 - 5(1) = 2 - 5 = -3. \]The point is \( (1, -3) \).

Step 4: Use point-slope form to find the equation of the tangent line

With the slope from Step 2, and the point from Step 3 \( (1, -3) \), use the point-slope form of a line: \[ y - y_1 = m(x - x_1) \]where \( m = 3 \), \( x_1 = 1 \), and \( y_1 = -3 \):\[ y + 3 = 3(x - 1). \]

Step 5: Convert the equation to standard form

Simplify the equation from Step 4 and convert it to the standard form by expanding and rearranging:\[ y + 3 = 3x - 3 \]\[ y = 3x - 6 \]Write it in standard form (Ax + By = C):\[ 3x - y = 6. \]

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that helps us understand how a function changes at any given point. Simply put, it measures the rate at which a function's output value changes concerning changes in its input value. When we talk about the tangent line to a curve, its slope at any point is given by the derivative.
To find the derivative of a function, we apply the rules of differentiation. In the context of our exercise, the function is given by \( y = 2x^4 - 5x \). Differentiating this function with respect to \( x \) gives us the derivative \( f'(x) = 8x^3 - 5 \).
Let's break down why knowing this derivative helps in finding the tangent line:

• The derivative tells us the exact steepness or slope of the curve at any point.
• For the tangent line at \( x = 1 \), we used the derivative to find \( f'(1) = 3 \), meaning the slope here is 3.

Understanding derivatives provides us a tool to not just measure change, but also predict behavior in various applications from physics to economics.

Point-Slope Form

The point-slope form is a valuable tool for writing the equation of a line when you're given a point on the line and the slope. It puts the complexity of a line equation into a manageable format. The general form of the point-slope equation is:
\[ y - y_1 = m(x - x_1) \]
Here:

• \( m \) is the slope of the line.
• \( (x_1, y_1) \) is the point through which the line passes.

In our solution, we identified the slope from the derivative as 3 and the point as \( (1, -3) \). Upon substituting these values into the point-slope formula, we derived the initial tangent line equation:
\[ y + 3 = 3(x - 1) \]
This format is particularly handy because:

• It clearly shows the slope and how the y-value changes as we move along the x-axis from the specific point \( (x_1, y_1) \).
• Provides a straightforward method to convert into other forms like the standard form.

Standard Form of a Line

The standard form of a line equation is a formal way of writing linear equations that has the format \( Ax + By = C \). Here, \( A \), \( B \), and \( C \) are integers, where \( A \) and \( B \) should not both be zero simultaneously.

Changing from point-slope to standard form helps present the equation in a neat and organized style, often preferred in mathematical and geometric scenarios. From the previous point-slope equation:
\[ y + 3 = 3(x - 1) \]
We expand and rearrange to simplify:

• First distribute: \( y + 3 = 3x - 3 \).
• Move all terms to one side to form: \( 3x - y = 6 \).

Adopting the standard form offers several advantages:

• Makes it easy to quickly identify and compare lines and their intercepts.
• Useful in solutions that require precise, integer-based calculations, such as certain graphing techniques or when integrating algebra problems.

Understanding how to switch between different forms of line equations enhances flexibility and depth in tackling mathematical problems.