Problem 1 - Calculus Single Variable

Question

Follow the outline given in this section to give a careful sketch of the graph of each of the functions in Exercises \(1-24\). Your sketch should exhibit, and have labeled, all of the following: a) local and global extrema, b) inflection points, c) intervals on which function is increasing or decreasing, d) intervals on which function is concave up or concave down, e) horizontal and vertical asymptotes. $$ f(x)=x^{3}-3 x^{2}-9 x+7 $$

Step-by-Step Solution

Answer

Graph has a local max at \(x = -1\), local min at \(x = 3\), inflection at \(x = 1\), increasing on \((-\infty,-1)\) and \((3,\infty)\), and no asymptotes.

Step 1: Find the first derivative of the function

To identify the critical points and determine intervals of increase or decrease, first find the first derivative of the function. The given function is \( f(x) = x^3 - 3x^2 - 9x + 7 \). The first derivative, \( f'(x) \), is obtained by differentiating each term: \[ f'(x) = 3x^2 - 6x - 9 \].

Step 2: Identify critical points

Critical points occur where \( f'(x) = 0 \) or \( f'(x) \) is undefined. Here \( f'(x) \) is never undefined. Set \( f'(x) = 0 \):\( 3x^2 - 6x - 9 = 0 \). Solve this quadratic equation using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \( a = 3 \), \( b = -6 \), and \( c = -9 \). This yields solutions \( x = 3 \) and \( x = -1 \).

Step 3: Determine intervals of increase/decrease

Use test points in the intervals around the critical points to determine the sign of \( f'(x) \):- For \( x < -1 \), choose \( x = -2 \): \( f'(-2) = 3(-2)^2 - 6(-2) - 9 = 3(4) + 12 - 9 = 15 > 0 \) (increasing).- For \( -1 < x < 3 \), choose \( x = 0 \): \( f'(0) = 3(0)^2 - 6(0) - 9 = -9 < 0 \) (decreasing).- For \( x > 3 \), choose \( x = 4 \): \( f'(4) = 3(4)^2 - 6(4) - 9 = 48 - 24 - 9 = 15 > 0 \) (increasing).

Step 4: Find the second derivative

Find the second derivative \( f''(x) \) to determine concavity and inflection points. Differentiate \( f'(x) \3x^2 - 6x - 9 \) to get \( f''(x) = 6x - 6 \).

Step 5: Find inflection points

Inflection points occur where \( f''(x) = 0 \) or changes sign. Set \( f''(x) = 0 \):\( 6x - 6 = 0 \), solve to get \( x = 1 \).Check for concavity change using test points:- For \( x < 1 \), choose \( x = 0 \): \( f''(0) = 6(0) - 6 = -6 < 0 \) (concave down).- For \( x > 1 \), choose \( x = 2 \): \( f''(2) = 6(2) - 6 = 6 > 0 \) (concave up).Thus, there is an inflection point at \( x = 1 \).

Step 6: Identify extrema and asymptotes

Local extrema correspond to changes in the sign of \( f'(x) \):- At \( x = -1 \), \( f(x) \) changes from increasing to decreasing (local maximum).- At \( x = 3 \), \( f(x) \) changes from decreasing to increasing (local minimum).Check function transformation for asymptotes. Since \( f(x) \) is a polynomial, it has no vertical or horizontal asymptotes due to its finite degree and lack of terms that make it undefined or 0 at infinitesimally large \( |x| \).

Step 7: Sketch the graph

With the information:- Extrema at \( x = -1 \) (local max) and \( x = 3 \) (local min).- Inflection point at \( x = 1 \).- Increasing on \(( -\infty, -1) \) and \((3, \infty) \), decreasing on \((-1, 3)\).- Concave up on \((1, \infty) \), concave down on \(( -\infty, 1) \).Plot these critical points, inflections, and intervals to sketch the general shape of the polynomial's curve.

Key Concepts

Graph Sketching

Graph sketching is a key skill in understanding the behavior of functions visually. For a given function like our polynomial, graph sketching involves identifying and marking key features such as critical points, intervals of increase and decrease, and points of inflection.
These features are crucial as they help predict the overall shape of the graph. To start sketching, make sure you have already calculated the derivatives, as this will guide you throughout the process. Remember to label significant points and clearly indicate different intervals on your graph.

• Begin by plotting the critical points and label them clearly on your graph.
• Identify and mark intervals where the function is increasing or decreasing.
• Add inflection points to your graph to show where the concavity changes.
• Ensure you represent all extrema, even if they are only local.

Using these guidelines, sketching the polynomial function can become an intuitive and straightforward task.

Derivatives

Derivatives are mathematical tools that help us understand how a function changes at any given point. They're essential for finding the slope of the function, which directly relates to its increase or decrease. When working with derivatives, start with the first derivative to identify critical points.
These critical points will indicate where the function might turn or flatten. For the polynomial function given, we computed the first derivative, leading to a quadratic expression. Solving for zero in this expression provides us the critical points.
Next, the second derivative offers insights into the function's concavity. A positive second derivative suggests the function is concave up, while a negative one indicates concave down. This information helps clarify the shape of the graph around the critical points and can also identify potential points of inflection.

Critical Points

Critical points are found where the first derivative of a function is zero or undefined. These points are where the function could have local minima or maxima. They're crucial in sketching graphs because they indicate points where the function might change direction.
To find critical points in our given polynomial function, we solved for when the first derivative equals zero. This led us to find two critical points at specific x-values. By evaluating the first derivative around these points, we determined where the function increased or decreased.

• If the derivative changes from positive to negative at a critical point, it's a local maximum.
• If the derivative changes from negative to positive, it's a local minimum.

This analysis is vital for obtaining an accurate graph rendering, as it indicates peaks and troughs in the graph of the function.

Concavity

Concavity describes the direction in which a curve bends. In calculus, it reveals whether a graph is arching upwards or hanging downwards. Understanding concavity helps anticipate the general shape of a function and determine potential points of inflection.
The second derivative tells us about the concavity of the function. If it's positive, the function is concave up (like a cup); if negative, it's concave down (like a frown). In our exercise, inflection points occur at zero crossings of the second derivative.
Evaluating the second derivative at points around a suspected inflection gives us concavity details. We found that for the polynomial function, the concavity changes at a specific x-value, indicating an inflection point. This switch in concavity affects the graph's sketch by altering the curve's direction.

Polynomial Functions

Polynomial functions are algebraic expressions made up of variables raised to any whole number power. They are characterized by their polynomial degrees, which influence the function's end behavior and number of possible turns or intersections.
In calculus, analyzing polynomial functions involves key tasks such as finding derivatives, critical points, and concavity to fully understand their graph. Our function here is a cubic polynomial, meaning it can intersect the x-axis up to three times and has a definite end behavior.

• The degree indicates the maximum number of turning points possible, which is always one less than the degree itself.
• Polynomials are smooth and continuous, making them straightforward to graph once key features are identified.
• Asymptotes are not found in typical polynomial graphs, as they are infinite in extent vertically and horizontally.

Understanding these properties helps simplify identifying the characteristics of the graph and ensures an accurate sketch.