Problem 45 - Brief Applied Calculus

Question

a. Find the equation for the tangent line to the curve \(f(x)=x^{2}-3 x+5\) at \(x=2\), writing the equation in slope-intercept form. [Hint: Use your answer to Exercise 25.] b. Use a graphing calculator to graph the curve together with the tangent line to verify your answer.

Step-by-Step Solution

Answer

The tangent line equation is \(y = x + 1\).

Step 1: Differentiate the Function

To find the equation of the tangent line, we first need to determine the slope at the given point. The slope of the tangent line is the derivative of the function evaluated at that point. Differentiate the function \(f(x) = x^2 - 3x + 5\) to find \(f'(x)\). Using the power rule, \(f'(x) = 2x - 3\).

Step 2: Evaluate the Derivative at the Given Point

Substitute \(x = 2\) into the derivative \(f'(x) = 2x - 3\) to find the slope of the tangent line at \(x = 2\). So, \(f'(2) = 2(2) - 3 = 4 - 3 = 1\). The slope of the tangent line is 1.

Step 3: Find the y-coordinate of the Point

Since we know the x-value, we need the corresponding y-value on the curve. Substitute \(x = 2\) into the original function \(f(x)\) to get the y-coordinate. \(f(2) = 2^2 - 3(2) + 5 = 4 - 6 + 5 = 3\). The point is (2, 3).

Step 4: Write the Equation of the Tangent Line

With the point (2, 3) and the slope 1, use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope. Substitute \(m = 1\), \(x_1 = 2\), and \(y_1 = 3\): \(y - 3 = 1(x - 2)\). Simplify to get the slope-intercept form: \(y = x + 1\).

Step 5: Verify with Graphing Calculator

Use a graphing calculator to plot the curve \(f(x) = x^2 - 3x + 5\) and the tangent line \(y = x + 1\). Verify visually that the line is indeed tangent to the curve at the point (2, 3). If both the curve and line meet solely at that point with the line just touching the curve and not crossing, the solution is verified.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that allows us to find the rate at which a function is changing at any given point. In simpler terms, it helps us determine the slope of a curve at any point, which is essential for finding tangent lines.

When you differentiate a function, you are essentially finding its derivative. The derivative provides a formula that tells you the slope of the tangent line at any point on the function's graph. By computing this, you can easily determine where the curve is increasing, decreasing, or remaining constant.

In the given exercise, we differentiated the function \( f(x) = x^2 - 3x + 5 \) using basic rules of differentiation. This process culminated in finding \( f'(x) = 2x - 3 \), which is the formula for the slope of the tangent line at any point \( x \) on the curve.

Slope-Intercept Form

Slope-Intercept Form is a linear equation format that is extremely helpful when working with linear functions and finding tangent lines. The formula is expressed as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept.

Knowing this form allows us to swiftly write down the equation of a line once we have the slope and a point through which the line passes. This form is not only easy to understand but also simplifies the graphing process, as you can readily determine where the line crosses the y-axis.

In our exercise, after calculating the slope \( m = 1 \) and finding the point (2, 3) on the curve, we plugged these values into the point-slope form and then rearranged it into slope-intercept form: \( y = x + 1 \). This gives us the equation of the tangent line in a clear and concise format.

Power Rule

The Power Rule is a key differentiation technique that's often used to find the derivative of polynomial functions. It states that if you have a function of the form \( ax^n \), its derivative is \( a \cdot n \cdot x^{n-1} \).

This rule is particularly useful because it's simple to apply and works for any real number exponent. It's one of the first rules learned in calculus for its straightforward application, especially for functions involving terms like \( x^2 \), \( x^3 \), or even fractions and negatives as exponents.

In our situation, to differentiate the function \( f(x) = x^2 - 3x + 5 \), we applied the Power Rule to each term:

• \( \frac{d}{dx}[x^2] = 2x \)
• \( \frac{d}{dx}[-3x] = -3 \)
• \( \frac{d}{dx}[5] = 0 \)

Collecting these, we obtained \( f'(x) = 2x - 3 \), which tells us the curve's slope at any point \( x \).

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing mathematical functions and verifying calculus computations, such as tangent lines. With just a few inputs, it can graph complex functions and overlay them with lines to check for intersections or tangential points.

By using a graphing calculator, you can visually confirm solutions obtained through algebraic methods. When a tangent line is graphed alongside its corresponding curve, you should observe it just touching the curve at a single point, without crossing it. This visual validation helps ensure mistakes are not made during manual computations.

In the exercise, verifying the solution involved plotting the original curve \( f(x) = x^2 - 3x + 5 \) alongside the tangent line \( y = x + 1 \). The graphing calculator helped confirm that these met precisely at the point (2, 3), solidifying our solution as correct.