Problem 39 - Calculus - AP Edition

Question

Determine whether the statement is true or false. Explain your answer. Find an equation of the tangent line to the graph of \(y=f(x)\) at \(x=-3\) if \(f(-3)=2\) and \(f^{\prime}(-3)=5\)

Step-by-Step Solution

Answer

True; the equation of the tangent line is \( y = 5x + 17 \).

Step 1: Understand the Problem

We need to find the equation of the tangent line to the function \( y = f(x) \) at the point where \( x = -3 \). We know \( f(-3) = 2 \), which gives us the y-coordinate and \( f^{\prime}(-3) = 5 \), which is the slope of the tangent line at that point.

Step 2: Recall the Point-Slope Form Formula

The point-slope form of the equation of a line is given by: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.

Step 3: Substitute Known Values into the Formula

We substitute \( m = 5 \), \( x_1 = -3 \), and \( y_1 = 2 \) into the point-slope equation: \[ y - 2 = 5(x + 3) \]

Step 4: Simplify the Equation

Distribute the \( 5 \) on the right side: \[ y - 2 = 5x + 15 \] Add \( 2 \) to both sides to solve for \( y \): \[ y = 5x + 17 \] This is the equation of the tangent line at \( x = -3 \).

Step 5: Determine the Truth of the Statement

The exercise asks if the statement about the equation of the tangent line is true or false. Since we calculated the equation of the tangent line using correct methods and values, the statement that this is the correct equation is true.

Key Concepts

Derivative

The concept of the derivative is fundamental in calculus. It represents the rate at which a function changes at any given point. Essentially, it describes the slope of the tangent line at a particular point on a graph. In our exercise, the derivative at the point \( x = -3 \) is given as \( f'(-3) = 5 \). This tells us that the slope of the tangent line to the curve at \( x = -3 \) is 5.

Understanding derivatives requires knowing that they are limits. They can be found by considering the difference quotient, which is the change in the function values divided by the change in input values, as the change approaches zero. Mathematically, if \( f(x) \) is a function, then its derivative \( f'(x) \) is given by the limit:

• \( f'(x) = \lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

Here, \( h \) is a small increment in \( x \). The derivative thus provides a powerful tool for analyzing and predicting how functions behave.

Point-Slope Form

Point-slope form is a method of writing the equation of a line, particularly useful when you know a slope and a point on the line. The point-slope formula is:

• \( y - y_1 = m(x - x_1) \)

where \( m \) is the slope and \( (x_1, y_1) \) is a specific point on the line.

In solving the exercise, knowing \( f'(-3) = 5 \) provided us with the slope \( m \). The coordinates \( (-3, 2) \) gave us the point \((x_1, y_1)\). By substituting these known values into the formula, we easily found the equation for our tangent line.

This form simplifies the process, especially in cases involving tangent lines, since we have a predefined point and the derivative provides us with an immediate slope. It’s efficient and direct, turning complex curvature into linear simplification at a point.

Calculus Problem Solving

Solving problems in calculus often involves a step-by-step thinking process, as demonstrated in the exercise. The first step is understanding what is given and what is required. Identifying key components such as the slope (derivative) and points on the function is crucial.

Once these are identified, choosing the correct formula or method is the next major step. For tangent lines, the point-slope form becomes essential because of its straightforward application. Substituting the known values into the equation can be done systematically:

• Identify the slope from the derivative.
• Use the point given on the curve.
• Apply these values to the point-slope form.

Simplifying the equation leads to the final solution. In our case, this methodology confirmed the equation \( y = 5x + 17 \) for the tangent line at \( x = -3 \), proving the statement true.

Understanding the broader context of calculus and its formulas allows not just for finding solutions, but for confirming their correctness, showing why calculus is integral in numeric reasoning.