Problem 43 - Calculus for Biology and Medicine

Question

In Problems , find the tangent line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=7 x^{3}+2 x-1, \text { at } x=-3 $$

Step-by-Step Solution

Answer

The tangent line at \(x = -3\) is \(191x - y = -377\).

Step 1: Find the Derivative

To find the tangent line, we first need to determine the slope at the point of tangency. The slope is given by the derivative of the function. The function is \[ y = f(x) = 7x^3 + 2x - 1 \].We find the derivative by using the power rule: \[ f'(x) = \frac{d}{dx}(7x^3 + 2x - 1) = 21x^2 + 2 \].

Step 2: Evaluate the Derivative at the Given Point

To find the slope of the tangent line at \( x = -3 \), we substitute \( -3 \) into the derivative:\[ f'(-3) = 21(-3)^2 + 2 = 21(9) + 2 = 189 + 2 = 191 \].Thus, the slope of the tangent line is 191.

Step 3: Find the Function Value at the Given Point

Next, find the y-coordinate of the tangent point by substituting \( x = -3 \) into the original function:\[ y = f(-3) = 7(-3)^3 + 2(-3) - 1 \].Calculate each term:\((-3)^3 = -27\), so \(7(-27) = -189\)\(2(-3) = -6\)Combine these with the constant:\[-189 - 6 - 1 = -196 \].The point of tangency is \((-3, -196)\).

Step 4: Write the Equation of the Tangent Line Using Point-Slope Form

With the point \((-3, -196)\) and slope 191, use the point-slope form: \[ y - y_1 = m(x - x_1) \].Substitute \(m = 191\), \(x_1 = -3\), and \(y_1 = -196\):\[ y + 196 = 191(x + 3) \].

Step 5: Convert to Standard Form

Rearrange the equation to the standard form \(Ax + By = C\):\[ y + 196 = 191x + 573 \].Subtract \(191x\) and \(573\) from both sides:\[ -191x + y = 377 \].Multiply by -1 to ensure \(A\) is positive (optional step):\[ 191x - y = -377 \].This is the standard form of the tangent line.

Key Concepts

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it. This means it just "grazes" the curve at that particular spot. Understanding tangent lines is crucial because they help us analyze how a function behaves at a specific point.

When finding the tangent line in calculus, we're essentially looking for two things:

• The point where the line touches the curve, which we call the point of tangency.
• The slope of this line at that very point, which reflects how steeply the curve is rising or falling.

In our original exercise, the function given is a cubic function. A cubic function typically has a curve that might change direction, making the ability to pinpoint the behavior at a single point particularly insightful.

The tangent line provides a simplified, linear approximation of the curve at the point of interest, which is essential in applications like physics and engineering, where we often estimate changes using tangents.

Derivative

The derivative of a function is a fundamental concept in calculus. It represents the rate at which a function's value changes as its input changes. In other words, it's the slope of the tangent line at any given point of the function. Calculating derivatives allows us to understand how a function behaves, providing insights into its increasing or decreasing nature.

In our exercise, the derivative helps us find the exact slope of the tangent line at a specific point on a cubic function. To find this derivative, we apply differentiation rules to the function, leading us to determine the slope at any given point.

By finding the derivative of the given function, we learn that:

• The derivative of a constant is zero.
• The derivative of a power of x follows the power rule, a foundational rule for differentiation.

This derivative information equips us with a critical tool not only for solving problems but also for analyzing real-world systems' dynamics over time.

Power Rule

The power rule is a fundamental concept in calculus for differentiating functions. It's a straightforward and powerful rule that simplifies finding the derivative of any function that involves powers of a variable. The power rule states that if you have a function of the form \(ax^n\), where \(a\) is a constant and \(n\) is a positive integer, its derivative is \(nax^{n-1}\).

In our specific exercise, the function involves terms like \(7x^3\), which fits perfectly for applying the power rule. Using the power rule, we determine the derivatives of each term:

• For \(7x^3\), the derivative is \(3 \times 7 \times x^{2} = 21x^2\).
• For the linear term \(2x\), the derivative is simply \(2\), as the power rule applies with \(n=1\).

These calculations lead to the derivative \(21x^2 + 2\), representing the slope function for our original cubic equation. Understanding this rule simplifies solving many calculus problems, making it an essential tool for students and professionals alike.