Problem 25 - Calculus of a Single Variable

Question

Finding an Equation of a Tangent Line In Exercises \(25-32,(\text { a) find an equation of the tangent line to the graph of } f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ f(x)=x^{2}+3, \quad(-1,4) $$

Step-by-Step Solution

Answer

The equation of the tangent line to the graph of \(f(x)=x^{2}+3\) at the point \((-1,4)\) is \(y = -2x + 2\).

Step 1: Obtain the Derivative

Find the derivative of the function \(f(x)=x^{2}+3\), which is \(f'(x) = 2x\). This derivative will be used to find the slope of the tangent line to the function at the given point.

Step 2: Substituting the Point into the Derivative

Substituting the x-coordinate of the given point \((-1,4)\) into the derivative \(f'(x) = 2x\), to calculate the slope of the tangent line at this point. Thus, the slope \(m\) of the tangent line is equal to \(f'(-1) = 2*(-1) = -2\).

Step 3: Use the Point-Slope Form to Find the Equation

Use the point-slope form of the equation of a line, which is \(y - y1 = m(x - x1)\), where \((x1, y1)\) are the coordinates of the given point, and \(m\) is the slope of the line. In this case, \(y - 4 = -2(x + 1)\). This can be simplified and rewritten in slope-intercept form to give the equation of the tangent line as \(y = -2x + 2\).

Step 4: Graph the Function and the Tangent Line

Use a graphing utility to graph the function \(f(x)=x^{2}+3\) and its tangent line \(y = -2x + 2\) at the point \((-1,4)\). The tangent line should touch the graph of the function at exactly this point.

Step 5: Confirm the Result Using a Derivative Feature

Use the derivative feature of a graphing utility to find the slope of the function at the point \((-1, 4)\). This value should match the slope of the tangent line calculated, which is -2, thus confirming the result.

Key Concepts

Derivative of a Function

When we speak of the derivative of a function, we are referring to a cornerstone of calculus—the rate at which a function is changing at any given point. In practical terms, it measures how fast the y-value (or the output) of a function changes as the x-value (or the input) changes. For the function in our exercise, f(x) = x^2 + 3, the derivative is denoted as f'(x) and is calculated using basic differentiation rules.

The derivative f'(x) of our quadratic function f(x) = x^2 + 3 is 2x, meaning for any x-value of the function, the rate of change or slope at that point is 2x. The derivative is paramount when determining the equation of a tangent line, as the slope of the tangent line at a specific point is precisely the value of the derivative at that point.

Point-Slope Form

To craft an equation for a tangent line, we can use the point-slope form of a linear equation, which is an incredibly useful tool in linear algebra. The point-slope form is given by (y - y1) = m(x - x1), where m is the slope of the line and (x1, y1) are the coordinates of a specific point on the line, typically the point of tangency on the graph.

In the context of our exercise, we determined the slope m of the tangent line at point (-1, 4) by evaluating the derivative f'(-1). After finding the slope to be -2, we plugged it into the point-slope form along with our point, yielding the equation y - 4 = -2(x + 1). This can then be transformed into other forms, like the slope-intercept form, for easier graphing and interpretation.

Graphing Utility Usage

Graphing utilities play a significant role in visualizing functions and their derivatives. They're an invaluable resource for confirming the accuracy of your work in calculus. You can input the original function f(x) = x^2 + 3, and the tangent line equation y = -2x + 2 to see whether the line indeed touches the curve at the given point (-1, 4).

Most graphing utilities also come equipped with a derivative feature. This feature allows you to directly calculate the slope of the function's tangent at any point, which should corroborate with your calculated slope—thus serving as a confirmation of your results. This step is like when we matched our manually found slope of the tangent line, -2, with the slope produced by the graphing utility when inputting the point (-1, 4).