Problem 35 - Calculus - AP Edition

Question

Find an equation for the line that is tangent to the curve \(y=x^{3}-2 x+1\) at the point \((0,1),\) and use a graphing utility to graph the curve and its tangent line on the same screen.

Step-by-Step Solution

Answer

The equation of the tangent line is \(y = -2x + 1\).

Step 1: Determine the derivative

First, we need to find the derivative of the function \(y = x^3 - 2x + 1\) since the derivative gives the slope of the tangent line. We apply the power rule to each term: \[y' = \frac{d}{dx}(x^3) - \frac{d}{dx}(2x) + \frac{d}{dx}(1).\]Simplifying each derivative gives us \[ y' = 3x^2 - 2. \]

Step 2: Evaluate the derivative at the given point

The tangent line at point \(0, 1\) requires the slope of the curve at that point. We find this by evaluating the derivative \(y'\) at \(x = 0\).\[y'(0) = 3(0)^2 - 2 = -2.\]

Step 3: Find the equation of the tangent line

The equation of a line can be found using the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \( (x_1, y_1) \) is the point of tangency. Using the slope from Step 2 and the point (0,1):\[ y - 1 = -2(x - 0) \]Which simplifies to:\[ y = -2x + 1. \]

Step 4: Graph the curve and tangent line

Use a graphing utility (such as Desmos or a graphing calculator) to plot the original function \(y = x^3 - 2x + 1\) and the tangent line \(y = -2x + 1\) on the same graph. You should see that the tangent line just touches the curve at the point \(0, 1\).

Key Concepts

Understanding the Derivative

The derivative of a function is a core concept in calculus. It is a fundamental tool used to calculate how a function's value changes with respect to changes in its input variables. To find the derivative of a function like \(y = x^3 - 2x + 1\), we apply differentiation rules to each term. In our case:

• For \(x^3\), the derivative is \(3x^2\) using the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\).
• For \(-2x\), the derivative is \(-2\), since the derivative of a constant multiplied by \(x\) is just the constant.
• Finally, the derivative of a constant like \(1\) is \(0\). Constant functions do not change, so their rate of change is zero.

Thus, combining these, we find \(y' = 3x^2 - 2\). This represents the slope of the tangent at any point \(x\) on the curve.

Slope of the Tangent Line

When we discuss the slope of the tangent line, we mean the inclination of the line that just touches the curve at a single point, without crossing it. This is crucial in finding tangent lines, as they reveal the instantaneous rate of change of the curve at that point. To find this slope for a function \(y = x^3 - 2x + 1\), we evaluate its derivative at the point of interest, which is at \(x = 0\). By substituting \(x = 0\) into the derivative \(y' = 3x^2 - 2\), we get \(y'(0) = 3(0)^2 - 2 = -2\). This means the slope of the tangent line at the point \((0, 1)\) is \(-2\). It's essential to grasp this idea, as the slope helps us write the equation of the tangent line in point-slope form.

Graphing Utility and Visualization

A graphing utility is a powerful tool that can visually confirm our analytical findings by plotting functions and their corresponding tangent lines. Using a graphing tool, like Desmos or a graphing calculator, you can input the original function \(y = x^3 - 2x + 1\) and the tangent line equation \(y = -2x + 1\) simultaneously. This visual representation allows you to see the curve and its tangent line plotted on the same screen. You will observe that the tangent line touches the curve exactly at the point \((0, 1)\), confirming that our calculated slope and tangent line equation are correct. Graphing utilities are invaluable for checking the accuracy of our calculus work and for improving our understanding of mathematical relationships by witnessing them visually.