Problem 52 - Calculus and its applications

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ f(x)=x^{2} ; \text { find } \lim _{x \rightarrow-1} f(x) \text { and } \lim _{x \rightarrow 0} f(x). $$

Step-by-Step Solution

Answer

\( \lim_{x \to -1} f(x) = 1 \); \( \lim_{x \to 0} f(x) = 0 \).

Step 1: Recognize the Function

The given function is a quadratic function, defined as \( f(x) = x^2 \). This is a basic parabola that opens upwards.

Step 2: Understanding Limits

Limits help us find the behavior of the function as \( x \) approaches a particular value. We must find \( \lim_{x \to -1} f(x) \) and \( \lim_{x \to 0} f(x) \).

Step 3: Calculate \( \lim_{x \to -1} f(x) \)

Substitute \( x = -1 \) into the function: \( f(x) = (-1)^2 = 1 \). Thus, \( \lim_{x \to -1} f(x) = 1 \).

Step 4: Calculate \( \lim_{x \to 0} f(x) \)

Substitute \( x = 0 \) into the function: \( f(x) = 0^2 = 0 \). Thus, \( \lim_{x \to 0} f(x) = 0 \).

Step 5: Graph the Function (Optional)

Plot the function \( f(x) = x^2 \) to visualize the parabola. It confirms the calculations that as \( x \to -1 \), \( f(x) \to 1 \) and as \( x \to 0 \), \( f(x) \to 0 \).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function, and it's one of the simplest yet most important functions in calculus. A quadratic function is typically written as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The presence of the squared term \( x^2 \) is what gives the quadratic function its unique characteristics.

Key features include:

• **Symmetry:** Quadratic functions are symmetric around a vertical line called the axis of symmetry. For \( f(x) = x^2 \), the axis of symmetry is simply \( x = 0 \).
• **Vertex:** This is the highest or lowest point on the graph of the quadratic function. In the function \( f(x) = x^2 \), the vertex is at the origin (0,0).
• **Opening:** The direction in which the parabola opens is determined by the sign of \( a \). If \( a > 0 \), the parabola opens upward, while if \( a < 0 \), it opens downward. For \( f(x) = x^2 \), it opens upwards since \( a = 1 \).

Understanding these features is crucial as they guide the characteristic shape of the function, which is a parabola.

Parabola

A parabola is the graph you obtain when you plot a quadratic function. In our specific example with the function \( f(x) = x^2 \), the parabola opens upwards and is centered at the origin. Here are some essential characteristics of parabolas:

• **Shape and Symmetry:** The shape is a smooth curve, with the direction of opening (up or down) indicated by the coefficient of the squared term. In the equation \( f(x) = x^2 \), the parabola is symmetric about the y-axis, meaning if you fold the parabola along the line \( x = 0 \), both sides match perfectly.
• **Graphing:** Visualizing a parabola helps in predicting the behavior of a quadratic function. You can plot points for various \( x \) values and connect them to form the curve. The vertex represents a special critical point, which is a minimum for upward-opening parabolas and a maximum for downward-opening ones.
• **Applications:** Parabolas model a host of real-world scenarios such as projectile motion, paths of celestial bodies, and even satellite dishes. Understanding their properties is key in fields such as physics and engineering.

Mastering the concept of parabolas makes it easier to analyze quadratic functions and solve practical problems that use their unique properties.

Limit Calculation

In calculus, calculating limits is a method used to understand the behavior of functions as they approach certain points. A limit is essentially the value a function "approaches" as the input (or \( x \)) gets closer to some number. Calculating the limits of \( f(x) = x^2 \) for different values shows us exactly how the function behaves near those points.

For example:

• **Limit as \( x \) approaches -1:** Substitute \( x = -1 \) into the function \( f(x) = (-1)^2 = 1 \). Thus, the limit \( \lim_{x \to -1} f(x) = 1 \). This means as \( x \) nears -1, \( f(x) \) gets closer to 1.
• **Limit as \( x \) approaches 0:** Substitute \( x = 0 \) into the function \( f(x) = 0^2 = 0 \). Thus, \( \lim_{x \to 0} f(x) = 0 \). So as \( x \) nears 0, \( f(x) \) gets closer to 0.

Limits are integral in understanding not just single points of the function, but also the overall continuity and behavior. They are fundamental to calculus and help in comprehending changes and approximations within various mathematical and real-world contexts.