Problem 1 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 8 , find the slope of the tangent line to the graph at the point \(\left(x_{1}, y_{1}\right) .\) Make a table of values of \(x, y\), and \(m\) at various points on the graph, and include in the table all points where the graph has a horizontal tangent. Draw a sketch of the graph. $$ y=9-x^{2} $$

Step-by-Step Solution

Answer

The slope is found using \(\frac{dy}{dx} = -2x\) and the horizontal tangent at (0, 9).

Step 1 - Derive the Function

First find the derivative of the function to determine the slope of the tangent line. The function is given as \(y = 9 - x^2\). Use derivative rules to find the first derivative. \[\frac{dy}{dx} = -2x\]

Step 2 - Find Slope at Specific Point

Substitute the given point \(x_1, y_1\) into the derivative to find the slope at that point. Let's find the slope at \(x = x_1, y = y_1\). For example, if \((x_1, y_1) = (1, 8)\), then the slope \(m\) is: \[m = \frac{dy}{dx}\bigg|_{x=1} = -2(1) = -2\]

Step 3 - Identify Horizontal Tangents

Set the derivative equal to zero to find where the function has horizontal tangents, because at these points, the slope of the tangent line is zero. \[-2x = 0\] Solve for \(x\). \[x = 0\]

Step 4 - Calculate y-values for Horizontal Tangents

Substitute the \(x\) values from Step 3 back into the original equation to find the corresponding \(y\) values. \[y = 9 - (0)^2 = 9\] Therefore, the point where the graph has a horizontal tangent is \( (0, 9) \).

Step 5 - Make the Table

Create a table that includes the \(x\), \(y\), and \(m\) values at various points. Include the points where the graph has a horizontal tangent.| x | y | m ||---|---|----|| 1 | 8 | -2 || 0 | 9 | 0 |

Step 6 - Sketch the Graph

Plot the function \( y = 9 - x^2 \) on a graph. Mark the points noted in the table and draw the tangent lines at these points. The graph is a downward-opening parabola with a vertex at (0,9).

Key Concepts

derivative

The derivative is a key concept in calculus. It measures how a function changes as its input changes. For instance, the derivative of a function gives us the slope of the tangent line to its graph at any given point. In our exercise, the function is given as \(y = 9 - x^2\). To find the derivative, we apply the rules of differentiation. The first derivative of this function is \[ \frac{dy}{dx} = -2x \] which indicates the slope of the tangent line at any point \(x\) on the graph.
To understand this better, let's consider a point \( (1, 8) \) on the graph. Substituting \(x = 1\) into the derivative, we get: \[ m = \frac{dy}{dx}\bigg|_{x=1} = -2(1) = -2 \].
So, at the point \( (1, 8) \), the slope of the tangent line is -2. This means the tangent line is decreasing steeply as we move from left to right.

horizontal tangent

A horizontal tangent line is a special case where the slope of the tangent line is zero. To find where a function has a horizontal tangent line, we set the derivative to zero and solve for \(x\). For our function \(y = 9 - x^2\), the derivative is \[ \frac{dy}{dx} = -2x \].
Setting the derivative equal to zero: \[ -2x = 0 \].
Solving for \(x\), we get \[ x = 0 \].
Next, we substitute \(x = 0\) back into the original equation to find the corresponding \(y\) value: \[ y = 9 - (0)^2 = 9 \].
Therefore, the function has a horizontal tangent at the point \( (0, 9) \).
A horizontal tangent means that at this point, the graph has a flat slope. It's essentially level at \( (0, 9) \), which is also the vertex of the parabola.

parabola

A parabola is a U-shaped curve that can open either upwards or downwards. In our exercise, we are dealing with the function \(y = 9 - x^2\), which graphs as a downward-opening parabola. This can be identified by the \(-x^2\) term, indicating that it opens downward because of the negative coefficient.
The vertex of this parabola is the highest point since it opens downwards. For the function \(y = 9 - x^2\), the vertex is at \((0, 9)\).To sketch the graph:

• Plot the vertex \((0, 9)\).
• Draw the curve opening downwards.
• Note that the parabola is symmetrical about the y-axis.

Additionally, it's useful to plot a couple more points and their tangents from the table of values, such as \((1, 8)\) with a slope of -2 and \((0, 9)\) with a horizontal tangent having a slope of 0.
Parabolas are common in many fields, including physics, where they often represent objects in freefall or trajectories in projectile motion.