Problem 18 - Calculus for Biology and Medicine

Question

Apply the product rule to find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=\left(3 x^{2}-2\right)(x-1)\), at \(x=1\)

Step-by-Step Solution

Answer

The equation of the tangent line is \(y = x - 1\).

Step 1: Recognize and Set Up the Product Rule

The product rule states that if you have a function defined as a product of two functions, \(f(x) = u(x)v(x)\), then its derivative is given by \(f'(x) = u'(x)v(x) + u(x)v'(x)\). For the function \(f(x) = (3x^2 - 2)(x - 1)\), identify \(u(x) = 3x^2 - 2\) and \(v(x) = x - 1\).

Step 2: Differentiate Each Part Using Basic Rules

Find the derivative of \(u(x)\) and \(v(x)\).- The derivative of \(u(x) = 3x^2 - 2\) is \(u'(x) = 6x\).- The derivative of \(v(x) = x - 1\) is \(v'(x) = 1\).

Step 3: Apply the Product Rule

Substitute the derivatives and functions back into the product rule:\[ f'(x) = (6x)(x - 1) + (3x^2 - 2)(1) \]Simplify:\[ f'(x) = 6x(x - 1) + 3x^2 - 2 \]\[ f'(x) = 6x^2 - 6x + 3x^2 - 2 \]\[ f'(x) = 9x^2 - 6x - 2 \].

Step 4: Evaluate the Derivative at the Point

To find the slope of the tangent line at \(x = 1\), substitute \(x = 1\) into \(f'(x)\):\[ f'(1) = 9(1)^2 - 6(1) - 2 = 9 - 6 - 2 = 1 \]Thus, the slope of the tangent line is 1.

Step 5: Find the Function Value at the Point

Find \(f(1)\) to determine the y-intercept of the tangent line:\[ f(1) = (3(1)^2 - 2)(1 - 1) = (3 - 2)(0) = 0 \].The point on the function is \((1, 0)\).

Step 6: Write the Equation of the Tangent Line

Use the point-slope form equation \(y - y_1 = m(x - x_1)\), where \(m = 1\) and the point is \((1,0)\).\[ y - 0 = 1(x - 1) \]Simplify to slope-intercept form:\[ y = x - 1 \].

Key Concepts

Tangent Line

A tangent line is an essential concept in calculus, and it refers to a straight line that just 'touches' a curve at a given point. This line represents the immediate direction of the curve at that specific point.

• Touching the Curve: The tangent line only meets the curve at one point, without crossing it.
• Slope: The slope of the tangent line at a specific point is critical as it indicates the steepness and direction the curve is heading right at that spot.

Understanding tangent lines helps us approximate the behavior of functions locally, and they are crucial in applications like calculating rates of change.
By finding the derivative of a function and substituting the specific point into it, we can determine the slope of the tangent line at that point. This slope is then used in equations to find the specific tangent line that represents the curve's behavior at that exact location.

Slope-Intercept Form

The slope-intercept form is a simple, standard way of expressing the equation of a straight line. It is written as:\[ y = mx + b \]where:

• \( m \) is the slope of the line, expressing how steep the line is.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

To form this equation from a problem, you need to first find the slope, usually through the derivative when working with calculus problems. After this, determine the point (x, y) where the line is tangent to the curve.
These values are then plugged into the line equation, often starting from the point-slope form \( y - y_1 = m(x - x_1) \). After simplifying, you transform it into the slope-intercept form. Slope-intercept form is intuitive and offers a quick visual interpretation of the line's behavior on a graph.

Derivative

A derivative is a fundamental concept in calculus and represents the rate of change of a function. It provides us with the slope of the function's graph at any given point.

• Basic Idea: Derivatives tell us how a function changes as its input changes.
• Notation: Often denoted as \( f'(x) \) or \( \frac{dy}{dx} \).

The derivative is calculated using rules like the product rule, chain rule, and quotient rule, depending on the function's complexity. For example, when dealing with a product of two functions, the product rule enables us to find the derivative:
"If \( f(x) = u(x)v(x) \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \)."
The derivative is the key tool we use to find tangent lines, as it gives the slope at a particular point. Additionally, understanding derivatives leads to more advanced applications such as finding maxima and minima of functions and solving real-world problems involving rates of change.